fluctuations, information, gravity and the quantum potential · fluctuations, information, gravity...

Fluctuations, Information, Gravity and the Quantum Potential

Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics:

Their Clarification, Development and Application

Editor:

ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board:

GIANCARLO GHIRARDI, University of Trieste, Italy

LAWRENCE P. HORWITZ, Tel-Aviv University, Israel

BRIAN D. JOSEPHSON, University of Cambridge, U.K.

CLIVE KILMISTER, University of London, U.K.

PEKKA J. LAHTI, University of Turku, Finland

ASHER PERES, Israel Institute of Technology, Israel

EDUARD PRUGOVECKI, University of Toronto, Canada

FRANCO SELLERI, Università di Bara, Italy

TONY SUDBERY, University of York, U.K.

HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der

Wissenschaften, Germany

Volume 148

Fluctuations, Information,Gravity and the Quantum Potential

by

R.W. CarrollUniversity of Illinois, Urbana, IL, U.S.A.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4003-2 (HB)

ISBN-10 1-4020-4025-3 (e-book)

Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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Printed in the Netherlands.

ISBN-13 978-1-4020-4003-0 (HB)

ISBN-13 978-1-4020-4025-2 (e-book)

© 2006 Springer

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Contents

ix

1. THE SCHRODINGER EQUATION 11. Diffusion and Stochastic Processes 12. Scale Relativity 143. Remarks on Fractal Spacetime 22

3.1. Comments on Cantor Sets 233.2. Comments on Hydrodynamics 27

4. Remarks on Fractal Calculus 305. A Bohmian Approach to Quantum Fractals 34

2. DEBROGLIE-BOHM IN VARIOUS CONTEXTS 391. The Klein-Gordon and Dirac Equations 41

1.1. Electromagnetism and the Dirac Equation 452. Bertoldi-Faraggi-Matone Theory 483. Field Theory Models 55

3.1. Emergence of Particles 553.2. Bosonic Bohmian Theory 613.3. Fermionic Theory 64

4. DeDonder, Weyl, and Bohm 715. 756. Bohmian Mechanics in QFT 87

3. GRAVITY AND THE QUANTUM POTENTIAL 911. Introduction 912. Sketch of Debroglie-Bohm-Weyl Theory 94

2.1. 1012.2. Remarks on Conformal Gravity 104

3. 1113.1. Fisher Information Revisited 1173.2. The KG Equation 119

4. Scale Relativity and KG 1265. Quantum Measurement and Geometry 132

5.1. Measurement on a Biconformal Space 137

4. GEOMETRY AND COSMOLOGY 1431. 1432. Remarks on Cosmology 151

v

Preface

QFT and Stochastic Jumps

Dirac-Weyl Action

The Schr dinger Equation in Weyl Space

Dirac-Weyl Geometry

o

vi CONTENTS

3. WDW Equation 1573.1. Constraints in Ashtekar Variables 162

4. Remarks on Regularization 1645. Pilot Wave Cosmology 170

5.1. Euclidean Quantum Gravity6. Bohm and Noncommutative Geometry 1777. Exact Uncertainty and Gravity 190

5. FLUCTUATIONS AND GEOMETRY 2031. The Zero Point Field 203

1.1. 2101.2. 2121.3. Massless Particles 2161.4. Einstein Aether Waves 218

2. 2213. Photons and EM 2274. Quantum Geometry 231

4.1. 237

6. INFORMATION AND ENTROPY 2391. The Dynamics of Uncertainty 239

1.1. Information Dynamics 2441.2. Information Measures for QM 2451.3. Phase Transitions 2461.4. Fisher Information and Hamilton’S Equations 2471.5. Uncertainty and Fluctuations 249

2. A Touch of Chaos 2522.1. 254

3. Generalized Thermostatistics 2563.1. 259

4. Fisher Physics 2624.1. Legendre Thermodynamics 2684.2. First and Second Laws 271

7. ON THE QUANTUM POTENTIAL 2751. Resume 275

1.1. The Schrodinger Equation 2751.2. Debroglie-Bohm 2791.3. Geometry, Gravity, and QM 2861.4. Geometric Phases 2911.5. Entropy and Chaos 293

2. Hydrodynamics and Geometry 2932.1. Particle and Wave Pictures 2962.2. 301

3. 3073.1. Discussion of a Putative Psi Aether 309

4. 315

172

Probability Aspects

Remarks on the Aether and Vacuum

Stochastic Electrodynamics

A Version of the Dirac Aether

Chaos and the Quantum Potential

Nonextensive Statistical Thermodynamics

Electromagnetism and the SE

Remarks on Trajectories

Some Speculations on the Aether

CONTENTS vii

8. REMARKS ON QFT AND TAU FUNCTIONS 3191. Introduction and Background 319

1.1. Wick’s Theorem and Renormalization 3211.2. Products and Relations to Physics 323

2. Quantum Fields and Quantum Groups 3273. Renormalization and Algebra 332

3.1. Various Algebras 3394. Tau Function and Free Fermions

4.1. Symmetric Functions 3434.2. PSDO on a Circle 347

5. Intertwining 3496. Vertex Operators and Symmetric Functions 358

APPENDIX A 365365

APPENDIX B 379Relativity and Electromagnetism 379

APPENDIX C 387Remarks on Quantum Gravity 387

APPENDIX D 395395

APPENDIX E 405Biconformal Geometry 405

Bibliography 411

Index 437

341

DeDonder-Weyl Theory

Dirac on Weyl Geometry

PREFACE

The quantum potential is a recurring theme and Leitmotiv for this book (alongwith variations on the Bohmian trajectory representation for quantum mechan-ics (QM)); it appears in many guises and creates connections between quantummechanics (QM), general relativity (GR), information, and Bohmian trajectories(as well as diffusion, thermodynamics, hydrodynamics, fractal structure, entropy,etc.). We make no claim to create quantum gravity for which there are severalembryonic theories presently under construction (cf. [53, 429, 783, 819, 829,896, 929]) and we will not deal with much of this beyond some use of the Ashtekarvariables for general relativity (cf. also [192, 228, 275, 258, 430, 460, 461, 462,578, 645, 744, 802, 803, 829, 902, 1012] for approaches involving posets, dis-crete differential manifolds, quantum causal spaces, groupoids, noncommutativegeometry, etc.). The main theme of the book is to relate ideas of quantum fluctua-tions (expressed via Fisher information for example, or diffusion processes, or frac-tal structures, or particle creation and annihilation, or whatever) in terms of the socalled quantum potential (which arises most conspicuously in deBroglie-Bohm the-ories). This quantum potential can be directly connected to the diffusion versionof the Schrodinger equation as in [672, 674, 698] and is also related to the Ricci-Weyl curvature of Dirac-Weyl theory ([186, 187, 188, 189, 219, 872, 873]). Re-cent work of A. and F. Shojai also develops a quantum potential and Schrodingerequation relative to the Wheeler-deWitt equation using Ashtekar variables (cf.[876]). Further, work of Israelit-Rosen on Weyl-Dirac theory leads to matter pro-duction by geometry (cf. [498, 499, 500, 501]) which could also be related to thequantum potential via the Dirac field as a matter field. In following this themewe have also been led to developments in scale relativity (cf. [715, 716, 718])where the Schrodinger equation arises from fractal structure and the quantumpotential clearly determines a quantization. Quantum potentials also arise in the(x, ψ) duality theme of Faraggi and Matone and we give a variation on thisrelated to the massless KG equation (or putative

The quantum fluctuation theme then leads also to stochasticelectrodynamics (SED) and the energy of the vacuum (zero point field - ZPF)and thence to further examination of electrodynamics, massless particles, etc. (cf.[190, 650, 753]). There is throughout the book an involvement with quantumfield theory (QFT) where in particular we extract from work of Nikolic, and there isconsiderable material devoted to entropy and information. In a sense the magicalstructure of quantum mechanics (QM) a la von Neumann and others is too per-fect; one cannot see what is “really” going on and this makes the deBroglie-Bohm

ix

et al. (cf. [1021]).aether equation) following Vancea

x PREFACE

theory attractive, where one has at least the illusion of having particle trajectories,etc. In fact we develop the theme that the Schrodinger mechanics of Hilbert spaceetc. is inherently incomplete (or uncertain); in particular the uncertainty can arisefrom its inability to see a third “initial” condition in the microstate trajectories(see Sections 2.2 and 7.4). We show in Remarks 2.2.2 and 3.2.1 how this producesexactly the Heisenberg uncertainty principle and how it is consequently naturalto consider ensembles of particles and probability densities for the Schrodingerpicture (cf. also [194, 197, 203, 346, 347, 373, 374]). The Hilbert space un-certainty is in fact automatic, due to the operator approach in Hilbert space, andis thus independent of quantum fluctuations, diffusion, fractals, microstates, orwhatever, so the Hilbert space formulation is indifferent to the interpretation ofuncertainty. However, we are interested in microcausality here and, furthermore,quantum fluctuations generating a quantum potential also have a direct relation touncertainty via the exact uncertainty principle of Hall-Reginatto; this is discussedat some length.

We want to make a few remarks about writing style. With a mathematicalbackground my writing style has acquired a certain flavor based on equations andalmost devoid of physical motivation or insights. I have tried to temper this whenwriting about physics but it is impossible to achieve the degree of physical insightcharacteristic of natural born physicists. I am comforted by the words of Dirac whoseemed to be guided and motivated by equations (although he of course possessedmore physical insight than I could possibly claim). In any case the “meaning” ofphysics for me lies largely in beautiful equations (Einstein, Maxwell, Schrodinger,Dirac, Hamilton-Jacobi, etc.) and the revelations about the universe presumablytherein contained. I hope to convey this spirit in writing and perhaps validatesomewhat this approach. Physics is a vast garden of delights and we can onlygape at some of the wonders (as expressed here via equations in directions of per-sonal esthetic appeal). We refrain from rashly suggesting that nature is simplya manifestation of underlying mathematical structure (i.e. symmetry, combina-torics, topology, geometry, etc.) played out on a global stage with energy andmatter as actors. On the other hand we are aware that much of physics, bothexperimental and theoretical, apparently has little to do with equations. Manyphenomena are now recognized as emergent (cf. [594, 860]) and one has to dealwith phase transitions, self organized criticality, chaos, etc. I know little aboutsuch matters and hope that the mathematical approach is not mistaken for arro-gance; it is only a dominant reverence for that beauty which I am able to perceive.The book of course is not finished; it probably cannot ever be finished since thereis new material appearing every weekday on the electronic bulletin boards. Hencewe have had to declare it finished as of April 10, 2005. It has reached the goal ofroughly 450 pages and includes whatever I conceived of as most important aboutthe quantum potential and Bohmian mechanics. I have learned a lot in writing thisand there is some original material (along with over 1000 references). Philosophyhas not been treated with much respect since I can only sense meaning in physicsthrough the equations. The fact that one can envision and manipulate “compos-ite” and abstract concepts or entities such as EM fields, energy, entropy, force,

PREFACE xi

gravity, mass, pressure, spacetime, spin, temperature, wave functions, etc. andthat there should turn out to be equations and relations among these “creatures”has always staggered my imagination. So does the fact that various combinationsof large and small numbers (such as c, e, G, , H etc.) can be combined intodimensionless form. Abstract mathematics has its own garden of concepts andrelations also of course but physics seems so much more real (even though I haveresisted any desire to experiment with these beautiful concepts, mainly because Iseem to have difficulty even in setting an alarm clock for example).

We will present a number of ideas which lack establishment deification, ei-ther because they are too new or, in heuristic approaches of considerable import,because of obvious conceptual flaws requiring futher consideration. There is alsosome speculative material, clearly labeled as such, which we hope will be produc-tive. The need for crazy ideas has been immortalized in a famous comment ofN. Bohr so there is no shame attached to exploratory ventures, however uncon-ventional they may seem at first appearance. The idea of a quantum potential asa link between classical and quantum phenomena, and the related attachment toBohmian type mechanics, seems compelling and is pursued throughout the book;that it should also have links to quantum fluctuations (and information) and Weylcurvature makes it irresistible.

There is no attempt to present a final version of anything. For a time in the1990’s the connection of string theory to soliton mathematics (e.g. in Seiberg-Witten theory) seemed destined to solve everything but it was only preliminary.Some of this is summarized nicely in a lovely paper of A. Morozov [662] whereconnections to matrix models and special functions (and much more) are indi-cated and the need for mathematical development in various directions is empha-sized (cf. [191, 662] and references there along with a few comments in Chap-ter 8). The arena of noncommutative geometry and quantum groups is person-ally very appealing (see e.g. [192, 258, 589, 615, 619, 620]) and we refer to[157, 158, 159, 160, 161, 162, 260, 580, 581, 582, 722] for some fascinatingwork on Feynman diagrams, quantum groups, and quantum field theory (some ofthis is sketched in Chapter 8). We have not gone into here (and know little about)several worlds of phenomenological physics regarding e.g. QCD, quarks, gluons,etc. (whose story is partly described in [997] by F. Wilczek in his Nobel lecture).We have also been fascinated by the mathematics of superfluids a la G. Volovik([968, 969]) and the mathematics is often related to other topics in this book;

There are also a fewremarks about cosmology and here again we know little and have only indicateda few places where Dirac-Weyl

play a role.

As to the book itself we mainly develop the theme of the quantum potentialand with it the Bohmian trajectory representation of QM. The quantum potentialarises most innocently in the Bohmian theory and the Schrodinger equation (SE)as an expression Q = −(2/2m)(∆|ψ|/|ψ|) where ψ is the wave function and Qappears then in the corresponding Hamilton-Jacobi (HJ) equation as a potential

however our understanding of the physics is very limited.

geometry, and hence perhaps the quantum potent-ial,

xii PREFACE

term. It is possible to relate this term to Fisher information, entropy, and quan-tum fluctuations in a natural manner and further to hydrodynamics, stress tensors,diffusion, Weyl-Ricci scalar curvatures, fractal velocities, osmotic pressure, etc. Itarises in relativistic form via Q = (2/m2c2))|ψ|/|ψ|) and in field theoretic mod-els as e.g. Q = −(2/2|Ψ|)(δ2|Ψ|/δ|Ψ|2). In terms of lapse and shift functions onehas a WDW version in the form Q = h2NqGijk(δ2|Ψ|/δqijδqk) (q is a surfacemetric). There is also a way in which the quantum potential can be considered asa mass generation term (cf. Remark 2.2.1 and [110]) and this is surely related toits role in Weyl-Dirac geometry in determining the matter field. In purely Weylgeometry one can use the quantum matter field (determined by Q) as a metricmultiplier to create the conformal geometry. In Chapter 7 we give a resume ofvarious aspects of the quantum potential contained in Chapters 1-7, followed byfurther information on the quantum potential. We had hoped to include materialon the mathematics of acoustics and superfluids a la Volovik and thermo-field-theory a la Blassone, Celeghini, Das, Iorio, Jizba, Rasetti, Vitiello, Umezawa, etal. (see e.g. [122, 231, 493, 519, 912, 913, 963, 964, 965]). This however istoo much (and no new insights into the quantum potential were visible); in anycase there are already books [286, 950, 968] available. Aside from a few shortsketches we have also felt that there is not enough room here to properly cover the

theory (QFT) at several places in the book, referring to [120,457, 935, 1015] for example, and we provide in Chapter 8 a survey articlewritten in 2003-2004 (QFT), tau

vertex operators. Although this does not touch onthe (and delves mainly into taming the combinatorics of QFTvia quantum groups for example), some of this material seems to be relevant tothe program suggested in

physics, with my sons David and Malcolm, and with my wife’s son Jim Bredt;I have also benefitted from

correspondence with C. Castro, M. Davidson, D. Delphenich, V. Dobrev, E. Floyd,G. Grossing, G. Kaniadakis, M. Kozlowski, M. Lapidus, M. Matone, H. Nikolic,

L. Bogdanov, F. Calogero, H. Doebner, J. Edelstein, P. Grinevich, Y. Kodama,B. Konopelchenko, A. Morozov, R. Parthasarathy, O. Pashaev, P. Sabatier, andM. Stone. A glance at the contents and index will indicate the important roleof H. Weyl in this book and it is perhaps appropriate to mention a “familial”

vatdozent. The book is dedicated to my wife Denise Rzewska-Bredt-Carroll whohas guided me in a passionate love affair through the perils of our golden years.

We have had recourseto quantum field

nonlinear Schrodinger equation (NLSE). (cf [223, 280, 292, 311, 312, 313, 413,691, 692, 693, 1028, 1029, 1030, 1031, 1032, 1033, 1034]).

(updated a bit) on quantum field theory func-

quantum potential

I would like to acknowledge interesting conversations about various topics in

andtions, Hopf algebras,

[662] and should be of current interest (cf. [191, 192]

F. Shojai, S. Tiwari, and J.T. Wheeler and from conversations with M. Bergvelt,

for more background). We recently became aware of important work in various

they all know much more than I about real physics.

1019, 1022, 1023, 1024] and cf. also [1020, 1021]).

influence as well, via my thesis advisor A. Weinstein who was Weyl’s only Pri-

directions by Brown, Hiley, deGosson, Padmanabhan, and Smolin (see [741, 1018

CHAPTER 1

THE SCHRODINGER EQUATION

Perhaps no subject has been the focus of as much mystery as “classical” quan-tum mechanics (QM) even though the standard Hilbert space framework providesan eminently satisfactory vehicle for determining accurate conclusions in manysituations. This and other classical viewpoints provide also seven decimal placeaccuracy in quantum electrodynamics (QED) for example. So why all the fuss?The erection of the Hilbert space edifice and the subsequent development of op-erator algebras (extending now into noncommutative (NC) geometry) has an airof magic. It works but exactly why it works and what it really represents remainshrouded in ambiguity. Also geometrical connections of QM and classical mechan-ics (CM) are still a source of new work and a modern paradigm focuses on theemergence of CM from QM (or below). Below could mean here a microstructureof space time, or quantum foam, or whatever. Hence we focus on other approachesto QM and will recall any needed Hilbert space ideas as they arise.

1. DIFFUSION AND STOCHASTIC PROCESSES

First consider the SE in the form −(2/2m)ψ′′ + V ψ = iψt so that forψ = Rexp(iS/) one obtains

(1.1) St +S2

X

2m+ V − 2R′′

2mR= 0; ∂t(R2) +

1m

(R2S′)′ = 0

1

mathematical and physical, and we sometimes avoid detailed technical discussionof mathematical fine points (cf. [241, 242, 243, 271, 315,

647, 672, 674, 698, 715, 719, 783, 810,

concerned with origins of the Schrodinger equation (SE). For background information

“structure”, both

726,

345, 531, 591, 592,607, 615, 672, 674, 810, 918] for various delicate matters).

There are some beautiful stochastic theories for diffusion and QM mainly

For example, ratherthan looking at such topics as Markov processes with jumps

with diffusion processes and kinetic theory.

we prefer to seek“meaning” for the Schrodinger equation via microstructure and

[33, 98, 131, 191, 192, 241, 242, 258, 471, 555,860, 1025, 1026,

1027]. The present development focuses on certain aspects of thediffusion processes,

quantum potentials, and fractalmethods. The aim is to envision

SE involving thewave function form ψ = Rexp(iS/), hydro dynamical versions,

615, 628,589,

591,

fractals inconnection

we mention for example

2 1. THE SCHRODINGER EQUATION

where S′ ∼ ∂S/∂X. Writing P = R2 (probability density ∼ |ψ|2) and Q =−(2/2m)(R′′/R) (quantum potential) this becomes

(1.2) St +(S′)2

2m+ Q + V = 0; Pt +

1m

(PS′)′ = 0

and this has some hydrodynamical interpretations in the spirit of Madelung. In-deed going to [294] for example we take p = S′ with p = mq for q a velocity (or“collective” velocity - unspecified). Then (1.2) can be written as (ρ = mP is anunspecified mass density)

(1.3) St +p2

2m+ Q + V = 0; Pt +

1m

(Pp)′ = 0; p = S′; P = R2;

Q = − 2

2m

R′′

R= − 2

2m

∂2√ρ√

ρ

Note here

(1.4)∂2√ρ√

ρ=

14

[2ρ′′

ρ−(

ρ′

ρ

)2]

Now from S′ = p = mq = mv one has

(1.5) Pt + (P q)′ = 0 ≡ ρt + (ρq)′ = 0; St +p2

2m+ V − 2

2m

∂2√ρ√

ρ= 0

Differentiating the second equation in X yields (∂ ∼ ∂/∂X, v = q)

(1.6) mvt + mvv′ + ∂V − 2

2m∂

(∂√

ρ√

ρ

)= 0

Consequently, multiplying by p = mv and ρ respectively in (1.5) and (1.6), weobtain

(1.7) mρvt + mρvv′ + ρ∂V − 2

2mρ∂

(∂2√ρ√

ρ

)= 0; mvρt + mv(ρ′v + ρv′) = 0

Then adding in (1.7) we get

(1.8) ∂t(ρv) + ∂(ρv2) +ρ

m∂V − 2

2m2ρ∂

(∂2√ρ√

ρ

)= 0

This is similar to an equation in [294] (called an “Euler” equation) and it definitelyhas a hydrodynamic flavor (cf. also [434] and see Section 6.2 for more details andsome expansion).

Now go to [743] and write (1.6) in the form (mv = p = S′)

(1.9)∂v

∂t+ (v · ∇)v = − 1

m∇(V + Q); vt + vv′ = −(1/m)∂(V + Q)

The higher dimensional form is not considered here but matters are similar there.This equation (and (1.8)) is incomplete as a hydrodynamical equation as a conse-quence of a missing term −ρ−1∇p where p is the pressure (cf. [607]). Hence one

1. DIFFUSION AND STOCHASTIC PROCESSES 3

“completes” the equation in the form

(1.10) m

(∂v

∂t+ (v · ∇)v

)= −∇(V + Q)−∇F ; mvt + mvv′ = −∂(V + Q)−F ′

where ∇F = (1/R2)∇p (or F ′ = (1/R2)p′). By the derivations above this wouldthen correspond to an extended SE of the form

(1.11) i∂ψ

∂t= − 2

2m∆ψ + V ψ + Fψ

provided one can determine F in terms of the wave function ψ. One notes that ita necessary condition here involves curlgrad(F ) = 0 or curl(R−2∇p) = 0 whichenables one to take e.g. p = −bR2 = −b|ψ|2. For one dimension one writes F ′ =−b(1/R2)∂|ψ|2 = −(2bR′/R)⇒ F = −2blog(R) = −blog(|ψ|2). Consequently onehas a corresponding SE

(1.12) i∂ψ

∂t= − 2

2mψ′′ + V ψ − b(log|ψ|2)ψ

This equation has a number of nice features discussed in [743]

For example ψ = βG(x−vt)exp(ikx−iωt) is a solution of (2.28) withV=0and forv= k/m one gets ψ = cexp[−(B/4)(x−vt+d)2] exp(ikx−iωt) where B = 4mb/2.Normalization

∫∞−∞ |ψ|

2 = 1 is possible with |ψ|2 = δm(ξ) =√

mα/πexp(−αmξ2)where α = 2b/2, d = 0, and ξ = x − vt F or m → ∞ we see that δm becomesa Dirac delta and this means that motionof alocalized. This is impossible for ordinary QM since exp(ikx − iωt) cannot belocalized as m→∞. Such behavior helps to explain the so-called collapse of thewave function and since superposition does not hold Schrodinger’s cat is eitherdead or alive. Further v = k/m is equivalent to the deBroglie relation λ = h/psince λ = (2π/k) = 2π(/mv) = 2π(h/2π)(1/p).

REMARK 1.1.1. We go now to [530] and the linear SE in the formi(∂ψ/∂t) = −(1/2m)∆ψ+U(r)ψ; such a situation leads to the Ehrenfest equationswhich have the form

(1.13) < v >= (d/dt) < r >; < r >=∫

d3x|ψ(r, t)|2r; m(d/dt) < v >=

= F (t)F (t) = −∫

d3x|ψ(r, t)|2∇U(r)

Thus the quantum expectation values of position and velocity of a suitable quan-tum system obey the classical equations of motion and the amplitude squared isa natural probability weight. The result tells us that besides the statistical fluc-tuations quantum systems posess an extra source of indeterminacy, regulated ina very definite manner by the complex wave function. The Ehrenfest theoremcan be extended to many point particle systems and in [530] one singles out thekind of nonlinearities that violate the Ehrenfest theorem. A theorem is provedthat connects Galilean invariance, and the existence of a Lagrangian whose Euler-Lagrange equation is the SE, to the fulfillment of the Ehrenfest theorem.

292,

particle with big mass is strongly

311, 312, 413, 691, 692, 693, 1028, 1029, 1030, 1031, 1032, 1033, 1034]).(cf. also [223, 280,


REMARK 1.1.2. There are many problems with the quantum mechanicaltheory of derived nonlinear SE (NLSE) but many examples of realistic NLSE arisein the study of superconductivity, Bose-Einstein condensates, stochastic modelsof quantum fluids, etc. and the subject demands further study. We make no at-tempt to survey this here but will give an interesting example later from [223]related to fractal structures where a number of the difficulties are resolved. Forfurther information on NLSE, in addition to the references above, we refer to[100, 281, 392, 413, 530, 534, 535, 536, 788, 789, 790, 956, 957] for sometypical situations (the list is not at all complete and we apologize for omissions).Let us mention a few cases.

• The program of [530] introduces a Schrodinger Lagrangian for a freeparticle including self-interactions of any nonlinear nature but no ex-plicit dependence on the space of time coordinates. The correspondingaction is then invariant under spatial coordinate transformations and byNoether’s theorem there arises a conserved current and the physical lawof conservation of linear momentum. The Lagrangian is also requiredto be a real scalar depending on the phase of the wave function onlythrough its derivatives. Phase transformations will then induce the lawof conservation of probability identified as the modulus squared of thewave function. Galilean invariance of the Lagrangian then determinesa connection between the probability current and the linear momentumwhich insures the validity of the Ehrenfest theorem.

• We turn next to [535] for a statistical origin for QM (cf. also [191, 281,534, 536, 698, 723, 809, 849]). The idea is to build a program inwhich the microscopic motion, underlying QM, is described by a rigorousdynamics different from Brownian motion (thus avoiding unnecessaryassumptions about the Brownian nature of the underlying dynamics).The Madelung approach gives rise to fluid dynamical type equations witha quantum potential, the latter being capable of interpretation in termsof a stress tensor of a quantum fluid. Thus one shows in [535] that thequantum state corresponds to a subquantum statistical ensemble whosetime evolution is governed by classical kinetics in the phase space. Theequations take the form

(1.14) ρt + ∂x(ρu) = 0; ∂t(µρui) + ∂j(ρφij) + ρ∂xiV = 0;

∂t(ρE) + ∂x(ρS)− ρ∂tV = 0

(1.15)∂S

∂t+

12µ

(∂S

∂x

)2

+W + V = 0

for two scalar fields ρ, S determining a quantum fluid. These can berewritten as

(1.16)∂ξ

∂t+

1µ

∂2S

∂x2+

1µ

∂ξ

∂x

∂S

∂x= 0;

∂S

∂t− η2

4µ

∂2ξ

∂x2− η2

8µ

(∂ξ

∂x

)2

+12µ

(∂S

∂x

)2

+ V = 0


where ξ = log(ρ) and for Ω = (ξ/2)+ (i/η)S = logΨ with m = Nµ, V =NV , and = Nη one arrives at a SE

(1.17) i∂Ψ∂t

= − 2

2m

∂2Ψ∂x2

+ VΨ

Further one can write Ψ = ρ1/2exp(iS/) with S = NS and here N =∫|Ψ|2dnx. The analysis is very interesting.

We will return to this later.

REMARK 1.1.3. Now in [324] one is obliged to use the form ψ = Rexp(iS/)to make sense out of the constructions (this is no problem with suitable provisos,e.g. that S is not constant - cf. [110, 191, 197, 198, 346, 347]). Thus noteψ′/ψ = (R′/R) + i(S′/) with (ψ′/ψ) = (1/m)S′ ∼ p/m (see also (1.22) below).Note also J = (/m)ψ∗ψ′ and ρ = R2 = |ψ|2 represent a current and a densityrespectively. Then using p = mv = mq one can write

(1.18) ′ J = (/m)|ψ|2(ψ∗ψ′/|ψ|2) = (/m)(ρ )

Then look at the SE in the form iψt = −(2/2m)ψ′′ + V ψ with ψt = (Rt +iStR/)exp(iS/) and

(1.19) ψxx = [(R′ + (iS′R/)exp(iS/)]′ =

[R′′ + (2iS′R′/) + (iS′′R/) + (iS′/)2R]exp(iS/)which means

(1.20) − 2

2m

[R′′ −

(S′

)2

+2iS′r′

+

iS′′R

]+ V R = i

[Rt +

iStR

]⇒

⇒ ∂tR2 +

1m

(R2S′)′ = 0; St +(S′)2

2mR− 2R′′

2mR+ V = 0

This can also be written as (cf. (1.3))

(1.21) ∂tρ +1m

∂(pρ) = 0; St +p2

2m+ Q + V = 0

where Q = −2R′′/2mR. Now we sketch the philosophy of [324, 325] in part.Most of such aspects are omitted here and we try to isolate the essential math-ematical features (see Section 1.2 for more). First one emphasizes configurationsbased on coordinates whose motion is choreographed by the SE according to therule (1-D only here)

(1.22) q = v =

mψ∗ψ′

|ψ|2

where iψt = −(2/2m)ψ′′ + V ψ. The argument for (1.22) is based on obtainingthe simplest Galilean and time reversal invariant form for velocity, transformingcorrectly under velocity boosts. This leads directly to (1.22) (cf. (1.18))) so thatBohmian mechanics (BM) is governed by (1.22) and the SE. It’s a fairly convincingargument and no recourse to Floydian time seems possible (cf. [191, 347, 373,374]). Note however that if S = c then q = v = (/m)(R′/R) = 0 whilep = S′ = 0 so perhaps this formulation avoids the S = 0 problems indicated in

p = (/m)(ψ /ψ); p

[191, 347, 373, 374]. One notes also that BM depends only on the Riemannianstructure g = (gij) = (miδij) in the form

(1.23) q = (gradψ/ψ); iψt = −(2/2)∆ψ + V ψ

What makes the constant /m in (1.22) important here is that with this value theprobability density |ψ|2 on configuration space is equivariant. This means that viathe evolution of probability densities ρt + div(vρ) = 0 (as in (1.21) with v ∼ p/m)the density ρ = |ψ|2 is stationary relative to ψ, i.e. ρ(t) retains the form |ψ(q, t)|2.One calls ρ = |ψ|2 the quantum equilibrium density (QED) and says that a systemis in quantum equilibrium when its coordinates are randomly distributed accord-ing to the QED. The quantum equilibrium hypothesis (QHP) is the assertion thatwhen a system has wave function ψ the distribution ρ of its coordinates satisfiesρ = |ψ|2.

REMARK 1.1.4. We extract here from [446, 447, 448] (cf. also the refer-ences there for background and [381, 382, 523] for some information geometry).There are a number of interesting results connecting uncertainty, Fisher informa-tion, and QM and we make no attempt to survey the matter. Thus first recall thatthe classical Fisher information associated with translations of a 1-D observableX with probability density P (x) is

(1.24) FX =∫

dx P (x)([log(P (x)]′)2 > 0

Recall now the Cramer-Rao inequality V ar(X) ≥ F−1X where V ar(X) ∼ variance

of X. A Fisher length for X is defined via δX = F−1/2X and this quantifies the length

scale over which p(x) (or better log(p(x))) varies appreciably. Then the root meansquare deviation ∆X satisfies ∆X ≥ δX. Let now P be the momentum observableconjugate to X, and Pcl a classical momentum observable corresponding to thestate ψ given via pcl(x) = (/2i)[(ψ′/ψ)− (ψ′/ψ)] (cf. (1.22)). One has then theidentity ψ=< pcl >ψ via integration by parts. Now define the nonclassicalmomentum by pnc = p−pcl and one shows that ∆X∆p ≥ δX∆p ≥ δX∆pnc = /2.Then go to [447] now where two proofs are given for the derivation of the SEfrom the exact uncertainty principle (δX∆pnc = /2). Thus consider a classicalensemble of n-dimensional particles of mass m moving under a potential V. Themotion can be described via the HJ and continuity equations

(1.25)∂s

∂t+

12m|∇s|2 + V = 0;

∂P

∂t+∇ ·

[P∇s

m

]= 0

for the momentum potential s and the position probability density P (note thatwe have interchanged p and P from [447] - note also there is no quantum potentialand this will be supplied by the information term). These equations follow fromthe variational principle δL = 0 with Lagrangian

(1.26) L =∫

dt dnx P

[∂s

∂t+

12m|∇s|2 + V

]It is now assumed that the classical Lagrangian must be modified due to theexistence of random momentum fluctuations. The nature of such fluctuations is

immaterial for (cf. [447] for discussion) and one can assume that the momentumassociated with position x is given by p = ∇s + N where the fluctuation term Nvanishes on average at each point x. Thus s changes to being an average momentumpotential. It follows that the average kinetic energy < |∇s|2 > /2m appearing in(1.26) should be replaced by < |∇s + N |2 > /2m giving rise to

(1.27) L′ = L + (2m)−1

∫dt < N ·N >= L + (2m)−1

∫dt(∆N)2

where ∆N =< N ·N >1/2 is a measure of the strength of the fluctuations. The ad-ditional term is specified uniquely, up to a multiplicative constant, by the followingthree assumptions

(1) Action principle: L′ is a scalar Lagrangian with respect to the fields Pand s where the principle δL′ = 0 yields causal equations of motion.Thus (∆N)2 =

∫dnx pf(P,∇P, ∂P/∂t, s,∇s, ∂s/∂t, x, t) for some scalar

function f .(2) Additivity: If the system comprises two independent noninteracting sub-

systems with P = P1P2 then the Lagrangian decomposes into additivesubsystem contributions; thus f = f1 + f2 for P = P1P2.

(3) Exact uncertainty: The strength of the momentum fluctuation at anygiven time is determined by and scales inversely with the uncertainty inposition at that time. Thus ∆N → k∆N for x → x/k. Moreover sinceposition uncertainty is entirely characterized by the probability densityP at any given time the function f cannot depend on s, nor explicitly ont, nor on ∂P/∂t.

The following theorem is then asserted (see [447] for the proofs).

THEOREM 1.1. The above 3 assumptions imply the relation (∆N)2 =c∫

dnx P |∇log(P )|2 where c is a positive universal constant.

COROLLARY 1.1. It follows from (1.27) that the equations of motion forp and s corresponding to the principle δL′ = 0 are

(1.28) i∂ψ

∂t= − 2

2m∇2ψ + V ψ

where = 2√

c and ψ =√

Pexp(is/).

REMARK 1.1.5. We sketch here for simplicity and clarity another deriva-tion of the SE along similar ideas following [805]. Let P (yi) be a probabilitydensity and P (yi + ∆yi) be the density resulting from a small change in the yi.Calculate the cross entropy via

(1.29) J(P (yi + ∆yi) : P (yi)) =∫

P (yi + ∆yi)logP (yi + ∆yi)

P (yi)dny

[12

∫1

P (yi)∂P (yi)

∂yi

∂P (yi)∂yk)

dny

]∆yi∆yk = Ijk∆yi∆yk


The Ijk are the elements of the Fisher information matrix. The most generalexpression has the form

(1.30) Ijk(θi) =12

∫1

P (xi|θi)∂P (xi|θi)

∂θj

∂P (xi|θi)∂θk

dnx

where P (xi|θi) is a probability distribution depending on parameters θi in additionto the xi. For P (xi|θi) = P (xi + θi) one recovers (1.29) (straightforward - cf.[805]). If P is defined over an n-dimensional manifold with positive inverse metricgik one obtains a natural definition of the information associated with P via

(1.31) I = gikIik =gik

2

∫1P

∂P

∂yi

∂P

∂ykdny

Now in the HJ formulation of classical mechanics the equation of motion takes theform

(1.32)∂S

∂t+

12gµν ∂S

∂xµ

∂S

∂xν+ V = 0

where gµν = diag(1/m, · · · , 1/m). The velocity field uµ is given by uµ = gµν(∂S/∂xν).When the exact coordinates are unknown one can describe the system by meansof a probability density P (t, xµ) with

∫Pdnx = 1 and

(1.33) (∂P/∂t) + (∂/∂xµ)(Pgµν(∂S/∂xν)) = 0

These equations completely describe the motion and can be derived from theLagrangian

(1.34) LCL =∫

P

∂S

∂t+

12gµν ∂S

∂xµ

∂S

∂xν+ V

dtdnx

using fixed endpoint variation in S and P. Quantization is obtained by adding aterm proportional to the information I defined in (1.31). This leads to(1.35)

LQM = LCL + λI =∫

P

∂S

∂t+

12gµν

[∂S

∂xµ

∂S

∂xν+

λ

P 2

∂P

∂xµ

∂P

∂xν

]+ V

dtdnx

Fixed endpoint variation in S leads again to (1.33) while variation in P leads to

(1.36)∂S

∂t+

12gµν

[∂S

∂xµ

∂S

∂xν+ λ

(1

P 2

∂P

∂xµ

∂P

∂xν− 2

P

∂2P

∂xµ∂xν

)]+ V = 0

These equations are equivalent to the SE if ψ =√

Pexp(iS/) with λ = (2)2.

REMARK 1.1.6. In Remarks 1.1.6 - 1.1.8 one uses Q = ±(1/m)times the standard Q = −(2/2m)(∆

√ρ/√

ρ. The SE gives to a probabilitydistribution ρ = |ψ|2 (with suitable normalization) and to this one can associatean information entropy S(t) (actually configuration information entropy) S =−∫

ρlog(ρ)d3x which is typically not a conserved quantity (S is an unfortunatenotation here but we retain it momentarily since no confusion should arise). The

rate of change in time of S can be readily found by using the continuity equationt

(1.37)∂S

∂t= −

∫ρt(1 + log(ρ))dx =

∫(1 + log(ρ))∂(vρ)

Note that a formal substitution of v = −u in the contintuity equation implies thestandard free Browian motion outcome dS/dt = D ·

∫[(∇ρ)2/ρ)d3x = D ·TrF ≥ 0

- use here u = D∇log(ρ) with D = /2m) and (1.37) with∫

(1 + log(ρ))∂(vρ) =−∫

vρ∂log(ρ) = −∫

vρ′ ∼∫

((ρ′)2/ρ) modulo constants involving D etc. Recallhere F ∼ −(2/D2)

∫ρQdx =

∫dx[(∇ρ)2/ρ] is a functional form of Fisher infor-

mation. A high rate of information entropy production corresponds to a rapidspreading (flattening down) of the probablity density. This delocalization featureis concomitant with the decay in time property quantifying the time rate at whichthe far from equilibrium system approaches its stationary state of equilibrium(d/dt)TrF ≤ 0.

REMARK 1.1.7. Now going back to the quantum context one admits gen-eral forms of the current velocity v. For example consider a gradient field v = b−uwhere the so-called forward drift b(x, t) of the stochastic process depends on aparticular diffusion model. Then one can rewrite the continuity equation as a

t Boundary restrictionsrequiring ρ, vρ, and bρ to vanish at spatial infinities or at boundaries yield thegeneral entropy balance equation

(1.38)dS

dt=∫ [

ρ(∇ · b) + D · (∇ρ)2

ρ

]d3x ≡ −D

dS

dt=∫

ρ(v · u)d3x =< v · u >

The first term in the first equation is not positive definite and can be interpretedas an entropy flux while the second term refers to the entropy production proper.The flux term represents the mean value of the drift field divergence ∇·b which byitself is a local measure of the flux incoming to or outgoing from an infinitesimalsurrounding of x at time t. If locally (∇· b)(x, t) > 0 on an infinitesimal time scalewe would encounter a local entropy increase in the system (increasing disorder)while in case (∇ · b)(x, t) < 0 one thinks of local entropy loss or restoration ororder. Only in the situation < ∇ · b >= 0 is there no entropy production. Quan-tum dynamics permits more complicated behavior. One looks first for a generalcriterion under which the information entropy S is a conserved quantity. Consider(1.8) and invoke the diffusion current to write (recall u = D(∇ρ)/ρ)

(1.39) DdS

dt= −

∫[ρ−1/2(ρv)] · [ρ−1/2(D∇ρ)]d3x

Then by means of the Schwarz inequality one has D|dS/dt| ≤< v2 >1/2< u2 >1/2

so a necessary (but insufficient) condition for dS/dt = 0 is that both < v2 >and < u2 > are nonvanishing. On the other hand a sufficient condition fordS/dt = 0 is that either one of these terms vanishes. Indeed in view of <u2 >= D2

∫[(∇ρ)2/ρ]d3x the vanishing information entropy production implies

dS/dt = 0; the vanishing diffusion current does the same job.

∂ ρ = −∇ · (vρ) where v is a current velocity field. Note here (cf. also [752])

standard Fokker-Planck equation ∂ ρ = D∆ρ − ∇ · (bρ).


REMARK 1.1.8. We develop a little more perspective now (following [395]- first paper). Recall Q written out as

(1.40) Q = 2D2 ∆ρ1/2

ρ1/2= D2

[∆ρ

ρ− 1

2ρ2(∇ρ)2

]=

12u2 + D∇ · u

where u = D∇log(ρ) is called an osmotic velocity field. The standard Brownianmotion involves v = −u, known as the diffusion current velocity and (up to adimensional factor) is identified with the thermodynamic force of diffusion whichdrives the irreversible process of matter exchange at the macroscopic level. On theother hand, even while the thermodynamic force is a concept of purely statisticalorigin associated with a collection of particles, in contrast to microscopic forceswhich have a direct impact on individual particles themselves, it is well knownthat this force manifests itself as a Newtonian type entry in local conservationlaws describing the momentum balance; in fact it pertains to the average (localaverage) momentum taken over by the particle cloud, a statistical ensemble prop-erty quantified in terms of the probability distribution at hand. It is precisely the(negative) gradient of the above potential Q in (1.40) which plays the Newtonianforce role in the momentum balance equations. The second analytical expressionof interest here involves

(1.41) −∫

Qρdx = (1/2)∫

u2ρdx = (1/2)D2 · FX ; FX =∫

(∇ρ)2

ρdx

where FX is the Fisher information, encoded in the probability density ρ whichquantifies its gradient content (sharpness plus localization/disorder) Note that

(1.42) −∫

Qρ = −∫

[(1/2)u2ρ + Dρu′] = −∫

(1/2)u2ρ +∫

Duρ′ =

= −(1/2)∫

D2(ρ′/ρ)2ρ + D2

∫ρ′(ρ′/ρ) = (D2/2)

∫(ρ′)2/ρ = (1/2)

∫u2ρ

On the other hand the local entropy production inside the system sustaining anirreversible process of diffusion is given via

(1.43)dS

dt= D ·

∫(∇ρ)2

ρdx = D · FX ≥ 0

This stands for an entropy production rate when the Fick law induced diffusioncurrent (standard Brownian motion case) j = −D∇ρ, obeying ∂tρ+∇j = 0, entersthe scene. Here S = −

∫ρlog(ρ)dx plays the role of (time dependent) information

entropy in the nonequilibrium statistical mechanics framework for the thermody-namics of irreversible processes. It is clear that a high rate of entropy increasecoresponds to a rapid spreading (flattening) of the probability density. This ex-plicitly depends on the sharpness of density gradients. The potential Q(x,t), theFisher information FX , the nonequilibrium measure of entropy production dS/dt,and the information entropy S(t) are thus mutually entangled quantities, eachbeing exclusively determined in terms of ρ and its derivatives.

−


In the standard statistical mechanics setting the Euler equation gives a pro-totypical momentum balance equation in the (local) mean

(1.44) (∂t + v · ∇)v =F

m− ∇P

ρ

where F = −∇F represents normal Newtonian force and P is a pressure term. Qappears in the hydrodynamical formalism of QM via

(1.45) (∂t + v · ∇)v =1m

F −∇Q =1m

F +2

2m2∇∆ρ1/2

ρ1/2

Another spectacular example pertains to the standard free Brownian motion inthe strong friction regime (Smoluchowski diffusion), namely

(1.46) (∂t + v · ∇)v = −2D2∇∆ρ1/2

ρ1/2= −∇Q

where v = −D(∇ρ/ρ) (formally D = /2m).

REMARK 1.1.9. The papers in [291, 292] contain very interesting deriva-tions of Schrodinger equations via diffusion ideas a la Nelson, Markov wave equa-tions, and suitable “applied” forces (e.g. radiative reactive forces).

We go now to Nagasawa [670, 671, 672, 673, 674] to see how diffusion andthe SE are really connected (cf. also [15, 141, 223, 421, 676, 681, 698, 726,732, 733, 734, 735, 736] for related material, some of which is discussed laterin detail); for now we simply sketch some formulas for a simple Euclidean met-ric where ∆ =

∑(∂/∂xi)2. Then ψ(t, x) = exp[R(t, x) + iS(t, x)] satisfies a SE

i∂tψ +(1/2)∆ψ + ia(t, x) ·∇ψ−V (t, x)ψ = 0 ( and m omitted with a(t, x) a driftcoefficient) if and only if

(1.47) V = −∂S

∂t+

12∆R +

12(∇R)2 − 1

2(∇S)2 − a · ∇S;

0 =∂R

∂t+

12∆S + (∇S) · (∇R) + a · ∇R

in the region D = (s, x) : ψ(s, x) = 0 (a harmless gauge factor in the divergenceis also being omitted). Solutions are often referred to as weak or distributionalbut we do not belabor this point. From [671, 672, 673] there results

THEOREM 1.2. Let ψ(t, x) = exp[R(t, x) + iS(t, x)] be a solution of theSE above; then φ(t, x) = exp[R(t, x) + S(t, x)] and φ = exp[R(t, x) − S(t, x)] aresolutions of

(1.48)∂φ

∂t+

12∆φ + a(t, x) · ∇φ + c(t, x, φ)φ = 0;

−∂φ

∂t+

12∆φ− a(t, x) · ∇φ + c(t, x, φ)φ = 0

where the creation and annihilation term c(t, x, φ) is given via

(1.49) c(t, x, φ) = −V (t, x)− 2∂S

∂t(t, x)− (∇S)2(t, x)− 2a · ∇S(t, x)


Conversely given (φ, φ) as in Theorem 1.2 satisfying (1.48) it follows that ψ satisfiesthe SE with V as in (1.49) (note R = (1/2)log(φφ) and S = (1/2)log(φ/φ) withexp(R) = (φφ)1/2).

We note that the equations (1.48) are not imaginary time SE and from allthis one can conclude that nonrelativistic QM is diffusion theory in terms ofSchrodinger processes (described by (φ, φ) - more details later). Further it isshown that certain key postulates in Nelson’s stochastic mechanics or Zambrini’sEuclidean QM (cf. [1011]) can both be avoided in connecting the SE to diffusionprocesses (since they are automatically valid). Look now at Theorem 1.2 for onedimension and write T = t with X = (/

√m)x and A = a/

√m; then the SE

becomes

(1.50) iψT = −(2/2m)ψXX − iAψX + V ψ;

iRT + (2/m2)RXSX + (2/2m2)SXX + ARX = 0;

V = −iST + (2/2m)RXX + (2/2m2)R2X − (2/2m2)S2

X −ASX

Hence

PROPOSITION 1.1. The SE of Theorem 1.2, written in the variables X =(/√

m)x, T = t, with A = (√

m/)a and V = V (X,T ) ∼ V (x, t) is equivalentto (2.2).

Making a change of variables in (1.48) now, as in Proposition 1.1, yields

COROLLARY 1.2. Equation (1.48), written in the variables of Proposition1.2, becomes

(1.51) φT +2

2mφXX + AφX + cφ = 0; −φT +

2

2mφXX −AφX + cφ = 0;

c = −V (X,T )− 2ST −2

mS2

X − 2ASX

Thus the diffusion processes pick up factors of and /√

m.

REMARK 1.1.10. We extract here from the Appendix to [672] for someremarks on competing points of view regarding diffusion and the the SE. Firstsome work of Fenyes [360] is cited where a Lagrangian is taken as

(1.52) L(t) =∫ [

∂S

∂t+

12(∇S)2 + V +

12

(12∇µ

µ

)2]

µdx

where µt(x) = exp(2R(t, x)) denotes the distribution density of a diffusion processand V is a potential function. The term Π(µ) = (1/2)[(1/2)(∇µ/µ)]2 is called adiffusion pressure and since (1/2)(∇µ/µ) ∼ ∇R the Lagrangian can be written as

(1.53) L =∫ [

∂S

∂t+

12(∇S)2 +

12(∇R)2 + V

]µdx


Applying the variational principle δ∫ b

aL(t)dt = 0 one arrives at

(1.54)∂S

∂t+

12

[(∇(R + S)]2− (∇(R+S)) ·(

12∇µ

µ

)+(

12∇µ

µ

)2

− 14

∆µ

µ+V = 0

which is called a motion equation of probability densities. From this he showsthat the function ψ = exp(R + iS) satisfies the SE i∂t + (1/2)∆ψ − V (t, x)ψ = 0.Indeed putting Π(µ) and the formula (1/2)(∆µ/µ)+(1/2)∆R+(∇R)2 into (1.53)one obtains

(1.55)∂S

∂t+

12(∇S)2 − 1

2(∇R)2 − 1

2∆R + V = 0

which goes along with the duality relation Rt + (1/2)∆S +∇S · ∇R + b · ∇R = 0where u = (1/2)(a + a) = ∇R and v = (1/2)(a − a) = ∇S as derived in theNagasawa theory. Hence ψ = exp(R + iS) satisfies the SE by previous calcula-tions. One can see however that the equation (1.53) is not needed since the SEand diffusion equations are equivalent and in fact the equations of motion are thediffusion equations. Moreover it is shown in [672] that (1.53) is an automaticconsequence in diffusion theory with V = −c− 2St − (∇S)2 and therefore it neednot be postulated or derived by other means. This is a simple calculation fromthe theory developed above.

REMARK 1.1.11. Nelson’s important work in stochastic mechanics [698]produced the SE from diffusion theory but involved a stochastic Newtonian equa-tion which is shown in [672] to be automatically true. Thus Nelson worked in ageneral context which for our purposes here can be considered in the context ofBrownian motions

(1.56) B(t) = ∂t + (1/2)∆ + b · ∇+ a · ∇; B(t) = −∂t + (1/2)∆− b · ∇+ a · ∇

and used a mean acceleration α(t, x) = −(1/2)[B(t)B(t)x+B(t)B(t)x]. Assumingthe duality relations after (1.55) he obtains a formula

(1.57) α(t, x) = −12[B(t)(−b+a)+B(b+a)] = bt+(1/2)∇(b)2−(b+v)×curl(b)−

−[−vt + (1/2)∆u + (1/2)(a · ∇)a + (1/2)(a · ∇)a− (b · ∇)v− (v · ∇)b− v× curl(b)]

Then it is shown that the SE can be deduced from the stochastic Newton’s equation

(1.58) α(t, x) = −∇V +∂b

∂t+

12∇(b2)− (b + v)× curl(b)

Nagasawa shows that this serves only to reproduce a known formula for V yieldingthe SE; he also shows that (1.57) also is an automatic consequence of the dualityformulation of diffusion equations above. This equation (1.57) is often called sto-chastic quantization since it leads to the SE and it is in fact correct with the Vspecified there. However the SE is more properly considered as following directlyfrom the diffusion equations in duality and is not correctly an equation of motion.There is another discussion of Euclidean QM developed by Zambrini [1011]. This

involves α(t, x) = (1/2)[B(t)B(t)x + B(t)B(t)x] (with (σσT )ij = δij). It is postu-lated that this equals −∇c + bt + (1/2)∇(b)2− b + v)× curl(b) which in fact leadsto the same equation for V as above with V = −c−2St− (∇S)2−2b ·∇S so thereis nothing new. Indeed it is shown in [672] that the postulated equivalence holdsautomatically as a simple consequence of time reversal of diffusion processes.

2. SCALE RELATIVITY

Scale relativity (SR) is due to L. Nottale (cf. [715, 716, 717, 718, 719, 720,721]) and somehow has not been accorded any real recognition by the “estab-lishment”. We only touch here on derivations of the SE and will develop furtheraspects later; the arguments are evidently heuristic but have a compelling inter-est. More general relativistic and cosmological features are discussed in Chapter2 where further discussion is given. The ideas involve spacetime having a fractalmicrostructure containing in particular continuous (self-similar) nondifferentiablepaths which serve as geodesic quantum paths of Hausdorff dimension D = 2. Thisis in fact a good notion of quantum path (following Feynman for example - cf.[1]) and we will see how it leads to a lovely (heuristic) derivation of the SE whichautomatically creates a complex wave function.

REMARK 1.2.1. One considers quantum paths a la Feynman so that(E1) limt→t′ [X(t) − X(t′)]2/(t − t′) exists. This implies X(t) ∈ H1/2 whereHα means cεα ≤ |X(t) − X(t′)| ≤ Cεα and from [345] for example this meansdimHX[a, b] = 1/2. Now one “knows” (see e.g. [1]) that quantum and Brown-ian motion paths (in the plane) have H-dimension 2 and some clarification isneeded here. We refer to [625] where there is a paper on Wiener Brownian motion(WBM), random walks, etc. discussing Hausdorff and other dimensions of varioussets. Thus given 0 < λ < 1/2 with probability 1 a Browian sample function Xsatisfies |X(t + h) − X(t)| ≤ b|h|λ for |h| ≤ h0 where b = b(λ). This leads tothe result that with probability 1 the graph of a Brownian sample function hasHausdorff and box dimension 3/2. On the other hand a Browian trail (or path) in2 dimensions has Hausdorff and box dimension 2 (note a quantum path can haveself intersections, etc.).

There are now several excellent approaches. The method of Nottale [700,715, 718] is preeminent (cf. also [732, 733, 734, 735]) and there is also a nicederivation of a nonlinear SE via fractal considerations in [223] (indicated below).The most elaborate and rigorous approach is due to Cresson [272], with elabo-ration and updating in [3, 273, 274]. There are various derivations of the SEand we follow [715] here (cf. also [718, 828]). The philosophy of scale relativitywill be discussed later and we just write down equations here pertaining to theSE. First a bivelocity structure is defined (recall that one is dealing with fractalpaths). One defines first

(2.1)d+

dty(t) = lim∆t→0+

⟨y(t + ∆t)− y(t)

∆t

⟩;

d−dt

y(t) = lim∆t→0+

⟨y(t)− y(t−∆t)

∆t

⟩

2. SCALE RELATIVITY 15

Applied to the position vector x this yields forward and backward mean velocities,namely (d+/dt)x(t) = b+ and (d−/dt)x(t) = b−. Here these velocities are definedas the average at a point q and time t of the respective velocities of the outgoingand incoming fractal trajectories; in stochastic QM this corresponds to an averageon the quantum state. The position vector x(t) is thus “assimilated” to a stochasticprocess which satisfies respectively after (dt > 0) and before (dt < 0) the instant ta relation dx(t) = b+[x(t)]dt+ dξ+(t) = b−[x(t)]dt+ dξ−(t) where ξ(t) is a Wienerprocess (cf. [698]). It is in the description of ξ that the D = 2 fractal characterof trajectories is inserted; indeed that ξ is a Wiener process means that the dξ’sare assumed to be Gaussian with mean 0, mutually independent, and such that

(2.2) < dξ+i(t)dξ+j(t) >= 2Dδijdt; < dξ−i(t)dξ−j(t) >= −2Dδijdt

where < > denotes averaging (D is now the diffusion coefficient). Nelson’s pos-tulate (cf. [698]) is that D = /2m and this has considerable justification (cf.[715]). Note also that (2.2) is indeed a consequence of fractal (Hausdorff) dimen-sion 2 of trajectories follows from < dξ2 > /dt2 = dt−1, i.e. precisely Feynman’sresult < v2 >1/2∼ δt−1/2 (the discussion here in [715] is unclear however - cf.[29]). Note also that Brownian motion (used in Nelson’s postulate) is known tobe of fractal (Hausdorff) dimension 2. Note also that any value of D may leadto QM and for D → 0 the theory becomes equivalent to the Bohm theory. Nowexpand any function f(x, t) in a Taylor series up to order 2, take averages, anduse properties of the Wiener process ξ to get

(2.3)d+f

dt= (∂t + b+ · ∇+D∆)f ;

d−f

dt= (∂t + b− · ∇ − D∆)f

Let ρ(x, t) be the probability density of x(t); it is known that for any Markov(hence Wiener) process one has ∂tρ + div(ρb+) = D∆ρ (forward equation) and∂tρ + div(ρb−) = −D∆ρ (backward equation). These are called Fokker-Planckequations and one defines two new average velocities V = (1/2)[b+ + b−] and U =(1/2)[b+−b−]. Consequently adding and subtracting one obtains ρt +div(ρV ) = 0(continuity equation) and div(ρU) − D∆ρ = 0 which is equivalent to div[ρ(U −D∇log(ρ))] = 0. One can show, using (2.3) that the term in square bracketsin the last equation is zero leading to U = D∇log(ρ). Now place oneself in the(U, V ) plane and write V = V − iU . Then write (dV/dt) = (1/2)(d+ + d−)/dtand (dU/dt) = (1/2)(d+ − d−)/dt. Combining the equations in (2.3) one defines(dV/dt) = ∂t + V · ∇ and (dU/dt) = D∆ + U · ∇; then define a complex operator(d′/dt) = (dV/dt)− i(dU/dt) which becomes

(2.4)d′

dt=(

∂

∂t− iD∆

)+ V · ∇

One now postulates that the passage from classical mechanics to a new nondif-ferentiable process considered here can be implemented by the unique prescriptionof replacing the standard d/dt by d′/dt. Thus consider S =

⟨∫ t2t1L(x,V, t)dt

⟩yielding by least action (d′/dt)(∂L/∂Vi) = ∂L/∂xi. Define then Pi = ∂L/∂Vi

leading to P = ∇S (recall the classical action principle with dS = pdq − Hdt).Now for Newtonian mechanics write L(x, v, t) = (1/2)mv2 − U which becomes


L(x,V, t) = (1/2)mV2 − U leading to −∇U = m(d′/dt)V. One separates real andimaginary parts of the complex acceleration γ = (d′V/dt to get

(2.5) d′V = (dV − idU )(V − iU) = (dVV − dUU)− i(dUV + dVU)

The force F = −∇U is real so the imaginary part of the complex accelerationvanishes; hence

(2.6)dUdt

V +dVdt

U =∂U

∂t+ U · ∇V + V · ∇U +D∆V = 0

from which ∂U/∂t may be obtained. This is a weak point in the derivation sinceone has to assume e.g. that U(x, t) has certain smoothness properties (see belowfor refinements). Differentiating the expression U = D∇log(ρ) and using thecontinuity equation yields another expression (∂U/∂t) = −D∇(divV )−∇(V ·U).Comparison of these relations yields ∇(divV ) = ∆V −U ∧ curlV where the curlUterm vanishes since U is a gradient. However in the Newtonian case P = mVso P∇S implies that V is a gradient and hence a generalization of the classicalaction S can be defined. Recall V = 2D∇S and ∇(divV ) = ∆V with curlV = 0;combining this with the expression for U one obtains S = log(ρ1/2) + iS. Onenotes that this is compatible with [698] for example. Finally set ψ =

√ρexp(iS) =

exp(iS) with V = −2iD∇(logψ) and note

(2.7) U = D∇log(ρ); V = 2D∇S;

V = −2iD∇logψ = −iD∇log(ρ) + 2D∇S = V − iU

Thus for P = mV the relation P ∼ −i∇ or Pψ = −i∇ψ has a naturalinterpretation. Putting ψ in the equation −∇U = m(d′/dt)V, which general-izes Newton’s law to fractal space the equation of motion takes the form ∇U =2iDm(d′/dt)(∇log(ψ)). Then noting that d′ and ∇ do not commute one replacesd′/dt by (2.4) to obtain

(2.8) ∇U = 2iDm [∂t∇log(ψ)− iD∆(∇log(ψ))− 2iD(∇log(ψ) · ∇)(∇log(ψ)]

This expression can be simplified via

(2.9) ∇∆ = ∆∇; (∇f · ∇)(∇f) = (1/2)∇(∇f)2;∆f

f= ∆log(f) + (∇log(f))2

which implies

(2.10)12∆(∇log(ψ)) + (∇log(ψ) · ∇)(∇log(ψ)) =

12∇∆ψ

ψ

Integrating this equation yields D2∆ψ + iD∂tψ− (U/2m)ψ = 0 up to an arbitraryphase factor α(t) which can be set equal to 0 by a suitable choice of phase S.Replacing D by /2m one arrives at the SE iψt = −(2/2m)∆ψ + Uψ and thissuggests an interpretation of QM as mechanics in a nondifferentiable (fractal)space.

In fact (using one space dimension for convenience) we see that if U = 0then the free motion m(d′/dt)V = 0 yields the SE iψt = −(2/2m)ψxx as ageodesic equation in “fractal” space. Further from U = (/m)(∂

√ρ/√

ρ) andQ = −(2/2m)(∆

√ρ/√

ρ) one arrives at a lovely relation, namely


PROPOSITION 2.1. The quantum potential Q can be written in the formQ = −(m/2)U2 − (/2)∂U (cf. (1.40) multiplied by −m). Hence the quantumpotential arises directly from the fractal nonsmooth nature of the quantum paths.Since Q can be thought of as a quantization of a classical motion we see that thequantization corresponds exactly to the existence of nonsmooth paths. Conse-quently smooth paths imply no quantum mechanics.

REMARK 1.2.2. In [15] (to be discussed later) one writes again ψ =Rexp(iS/) with field equations in the hydrodynamical picture (1-D for conve-nience)

(2.11) dt(m0ρv) = ∂t(m0ρv) +∇(m0ρv) = −ρ∇(u + Q); ∂tρ +∇ · (ρv) = 0

where Q = −(2/2m0)(∆√

ρ/√

ρ). The Nottale approach is used as above withdv ∼ dV and du ∼ dU . One assumes that the velocity field from the hydrody-namical model agrees with the real part v of the complex velocity V = v − iu sov = (1/m0)∇s ∼ 2D∂s and u = −(1/m0)∇σ ∼ D∂log(ρ) where D = /2m0. Inthis context the quantum potential Q = −(2/2m0)∆D

√ρ/√

ρ becomes

(2.12) Q = −m0D∇ · u− (1/2)m0u2 ∼ −(/2)∂u− (1/2)m0u

2

Consequently Q arises from the fractal derivative and the nondifferentiability ofspacetime again, as in Proposition 2.1. Further one can relate u (and hence Q) toan internal stress tensor whereas the v equations correspond to systems of Navier-Stokes type.

REMARK 1.2.3. Some of the relevant equations for dimension one arecollected together later. We note that it is the presence of ± derivatives that makespossible the introduction of a complex plane to describe velocities and hence QM;one can think of this as the motivation for a complex valued wave function andthe nature of the SE.

We go now to [223] and will sketch some of the material. Here one extendsideas of Nottale and Ord in order to derive a nonlinear Schrodinger equation(NLSE). Using the hydrodynamic model in [743] one added a hydrostatic pressureterm to the Euler-Lagrange equations and another possibility is to add instead akinematic pressure term. The hydrostatic pressure is based on an Euler equation−∇p = ρg where ρ is density and g the gravitational acceleration (note this gives

p = ρgx in 1-D). In [743] one took ρ = ψ∗ψ, b a mass-energy parameter, andp = ρ; then the hydrostatic potential is (for ρ0 = 1)

(2.13) b

∫g(x) · dr = −b

∫ ∇p

ρ· dr = −blog(ρ/ρ0) = −blog(ψ∗ψ)

Here −blog(ψ∗ψ) has energy units and explains the nonlinear term of [111] whichinvolved

(2.14) i∂ψ

∂t= − 2

2m∇2ψ + Uψ − b[log(ψ∗ψ)]ψ

A derivation of this equation from the Nelson stochastic QM was given by Lemos

−

(cf. [588]). There are moreover some problems since this equation does not obey thehomogeneity condition saying that the state λ|ψ > is equivalent to |ψ >; moreover

−

(2.14) is not invariant under ψ → λψ. Further, plane wave solutions to (2.14) donot seem to have a physical interpretion due to extraneous dispersion relations.Finally one would like to have a SE in terms of ψ alone. Note that another NLSEcould be obtained by adding kinetic pressure terms (1/2)ρv2 and taking ρ = aψ∗ψwhere v = p/m. Now using the relations from HJ theory (ψ/ψ∗) = exp[2iS(x)/]and p = ∇S(x) = mv one can write v = −i(/2m)∇log(ψ/ψ∗) so that the energydensity becomes

(2.15) (1/2)ρ|v|2 = (a2/8m2)ψψ∗∇log(ψ/ψ∗) · ∇log(ψ∗/ψ)

This leads to a corresponding nonlinear potential associated with the kinematicalpressure via (a2/8m2)∇log(ψ/ψ∗) · ∇log(ψ∗/ψ). Hence a candidate NLSE is

(2.16) i∂t = − 2

2m∇2ψ + Uψ − b[log(ψ∗ψ)]ψ +

a2

8m2

(∇log

ψ

ψ∗ · ∇logψ∗

ψ

)Here the Hamiltonian is Hermitian and a = b are both mass-energy parametersto be determined experimentally. The new term can also be written in the form∇log(ψ/ψ∗) · ∇log(ψ∗/ψ) = −[∇log(ψ/ψ∗)]2. The goal now is to derive a NLSEdirectly from fractal space time dynamics for a particle undergoing Brownian mo-tion. This does not require a quantum potential, a hydrodynamic model, or anypressure terms as above.

REMARK 1.2.4. One should make some comments about the kinematicpressure terms (1/2)ρv2 ∼ (2/2m)(a/m)|∇log(ψ)|2 versus hydrostatic pressureterms of the form

∫(∇p/ρ) ∼ −blog(ψ∗ψ). The hydrostatic term breaks homo-

geneity whereas the kinematic pressure term preserves homogeneity (scaling witha λ factor). The hydrostatic pressure term is also not compatible with the motionkinematics of a particle executing a fractal Brownian motion. The fractal formu-lation will enable one to relate the parameters a, b to .

Following Nottale, nondifferentiability implies a loss of causality and one isthinking of Feynmann paths with < v2 >∝ (dx/dt)2 ∝ dt2[(1/D)−1) with D = 2.Now a fractal function f(x, ε) could have a derivative ∂f/∂ε and renormalizationgroup arguments lead to (∂f(x, ε)/∂logε) = a(x) + bf(x, ε) (cf. [715]). This canbe integrated to give f(x, ε) = f0(x)[1 − ζ(x)(λ/ε)−b]. Here λ−bζ(x) is an inte-gration constant and f0(x) = −a(x)/b. This says that any fractal function can beapproximated by the sum of two terms, one independent of the resolution and theother resolution dependent; ζ(x) is expected to be a flucuating function with zeromean. Provided a = 0 and b < 0 one has two interesting cases (i) ε << λ withf(x, ε) ∼ f0(x)(λ/ε)−b and (ii) ε >> λ with f independent of scale. Here λ is thedeBroglie wavelength. Now one writes

(2.17) r(t + dt, dt)− r(t, dt) = b+(r, t)dt + ξ+(t, dt)(

dt

τ0

)β

;

r(t, dt)− r(t− dt, dt)− b−(r, t)dt + ξ−(t, dt)(

dt

τ0

)β

where β = 1/D and b± are average forward and backward velocities. This leadsto v±(r, t, dt) = b±(r, t) + ξ±(t, dt)(dt/τ0)β−1. In the quantum case D = 2 one

has β = 1/2 so dtβ−1 is a divergent quantity (i.e. nondifferentiability ensues).Following [588, 715, 698] one defines

(2.18)d±r(t)

dt= lim∆t→±0

⟨r(t + ∆t)− r(t)

∆t

⟩from which d±r(t)/dt = b±. Now following Nottale one writes

(2.19)δ

dt=

12

(d+

dt+

d−dt

)− i

2

(d+

dt− d−

dt

)which leads to (δ/dt) = (∂/∂t)+v ·∇− iD∇2. Here in principle D is a real valueddiffusion constant to be related to . and < dξ±idξ±j >= ±2Dδijdt. Now for thecomplex time dependent wave function we take ψ = exp[iS/2mD] with p = ∇S sothat v = −2iD∇log(ψ). The SE is obtained from the Newton equation (F = ma)via −∇U = m(δ/dt)v = −2imD(δ/dt)∇log(ψ) which yields

(2.20) −∇U = −2im[D∂t∇log(ψ)]− 2D∇(D∇

2ψ

ψ

)(see [715] for identities involving∇). Integrating yieldsD2∇2ψ+iD∂tψ−(U/2m)ψ =0 up to an arbitrary phase factor which may be set equal to zero. Now replacingD by /2m one gets the SE i∂tψ + (2/2m)∇2ψ = Uψ. Here the Hamiltonianis Hermitian, the equation is linear, and the equation is homogeneous of degree 1under the substitution ψ → λψ.

Next one generalizes this by relaxing the assumption that the diffusion co-efficient is real. Some comments on complex energies are needed - in particularconstraints are often needed (cf. [788]). However complex energies are not alienin ordinary QM (cf. [223] for references). Now the imaginary part of the linearSE yields the continuity equation ∂tρ +∇ · (ρv) = 0 and with a complex potentialthe imaginary part of the potential will act as a source term in the continuityequation. Instead of < dζ±dζ± >= ±2Ddt with D and 2mD = real one sets

(2.21) < dζ±dζ± >= ±(D +D∗)dt; 2mD = = α + iβ

The complex time derivative operator becomes (δ/dt) = ∂t+v·∇−(i/2)(D+D∗)∇2.Writing again ψ = exp[iS/2mD] = exp(iS/) one obtains v = −2iD∇log(ψ).The NLSE is then obtained (via the Newton law) via the relation−∇U = m(δ/dt)v =−2imD(δ/dt)∇log(ψ). Combining equations yields then

(2.22) ∇U = 2im[D∂t∇log(ψ)− 2iD2(∇log(ψ) · ∇)(∇log(ψ)−

− i

2(D +D∗)D∇2(∇log(ψ)]

Now using the identities (i)∇∇2 = ∇2∇, (ii) 2(∇log(ψ)·∇)(∇log(ψ) = ∇(∇log(ψ))2

and (iii) ∇2log(ψ) = ∇2ψ/ψ − (∇log(ψ))2 leads to a NLSE with nonlinear (kine-matic pressure) potential, namely

(2.23) i∂tψ = − 2

2m

α

∇2ψ + Uψ − i

2

2m

β

(∇log(ψ))2ψ

Note the crucial minus sign in front of the kinematic pressure term and also that = α + iβ = 2mD is complex. When β = 0 one recovers the linear SE. The

nonlinear potential is complex and one defines W = −(2/2m)(β/)(∇log(ψ))2

with U the ordinary potential; then the NLSE is

(2.24) i∂tψ = [−(2/2m)(α/)∇2 + U + iW ]ψ

This is the fundamental result of [223]; it has the form of an ordinary SE withcomplex potential U +iW and complex . The Hamiltonian is no longer Hermitianand the potential itself depends on ψ. Nevertheless one can have meaningful phys-ical solutions with real valued energies and momenta; the homogeneity breakinghydrostatic pressure term −b(log(ψ∗ψ)ψ is not present (it would be meaningless)and the NLSE is invariant under ψ → λψ.

REMARK 1.2.5. One could ask why not simply propose as a valid NLSEan equation

i∂tψ = − 2

2m∇2ψ +

2

2m

a

m|∇log(ψ)|2ψ

Here one has a real Hamiltonian satisfying the homogeneity condition and theequation admits soliton solutions of the form ψ = CA(x − vt)exp[i(kx − ωt)]where A(x − vt) is to be determined by solving the NLSE. The problem here isthat the equation suffers from an extraneous dispersion relation. Thus puttingin the plane wave solution ψ ∼ exp[−i(Et − px)] one gets an extraneous energymomentum (EM) relation (after setting U = 0), namely E = (p2/2m)[1 + (a/m)]instead of the usual E = p2/2m and hence EQM = EFT where FT means fieldtheory.

REMARK 1.2.6. It has been known since e.g. [788] that the expressionfor the energy functional in nonlinear QM does not coincide with the QM energyfunctional, nor is it unique. To see this write down the NLSE of [111] in the formi∂tψ = ∂H(ψ, ψ∗)/∂ψ∗ where the real Hamiltonian density is

H(ψ, ψ∗) = − 2

2mψ∗∇2ψ + Uψ∗ψ − bψ∗log(ψ∗ψ)ψ + bψ∗ψ

Then using EFT =∫

Hd3r we see it is different from < H >QM and in factEFT − EQM =

∫bψ∗ψd3r = b. This problem does not occur in the fractal based

NLSE since it is written entirely in terms of ψ.

REMARK 1.2.7. In the fractal based NLSE there is no discrepancy betweenthe QM energy functional and the FT energy functional. Both are given by

NNLSEfractal = − 2

2m

α

ψ∗∇2ψ + Uψ∗ψ − i

2

2m

β

ψ∗(∇log(ψ)2ψ

The NLSE is unambiguously given by in Remark 1.2.5 and H(ψ, ψ∗) is homoge-neous of degree 1 in λ. Such equations admit plane wave solutions with dispersionrelation E = p2/2m; indeed, inserting the plane wave solution into the fractalbased NLSE one gets (after setting U = 0)

(2.25) E =2

2m

α

p2

2m+ i

β

p2

2m=

p2

2m

α + iβ

=

p2

2m

since = α + iβ. The remarkable feature of the fractal approach versus allother NLSE considered sofar is that the QM energy functional is precisely the FTone. The complex diffusion constant represents a truly new physical phenomenoninsofar as a small imaginary correction to the Planck constant is the hallmark ofnonlinearity in QM (see [223] for more on this).

REMARK 1.2.8. Some refinements of the Nottale derivation are given in[272] and we consider x → f(x(t), t) ∈ Cn with X(t) ∈ H1/n (i.e. cε1/n ≤|X(t′)−X(t)| ≤ Cε1/n). Define (f real valued)

(2.26) ∇ε±f(t) =

f(t± ε)− f(t)±ε

;εf

t(f) =

12(∇ε

+ +∇ε−)f − i

2(∇ε

+ −∇ε−f);

aε,j(t) =12[(∆ε

+x)j − (−1)j(∆ε−x)j ]− i

2[(∆ε

+x)j + (−1)j(∆ε−x)j ]

Here one assumes h > 0 and ε(f, h) ≥ ε > 0 where ε(f, h) is the minimal resolutiondefined via infεaε(f) < h for aεf(t) = |[f(t+ε)+f(t−ε)−2f(t)]/ε|. If ε(f, h) isnot 0 then f is not differentiable (but not conversely). Now assume some minimalcontrol over the lack of differentiability (cf. [272]) and then for f now complexvalued with εf/t = (εf/t) + i(εf/t) (note the mixing of i terms is nottrivial) one has

(2.27)εf

t=

∂f

∂t+

εx

t

∂f

∂x+

n∑2

1j!

aε,j(t)∂jf

∂xjεj−1 + o(ε1/n)

We sketch now the derivation of a SE in the spirit of Nottale but with moremathematical polish. Going to [272] one defines (for a nondifferentiable functionf)

(2.28) fε(t) =12ε

∫ t+ε

t−ε

f(s)ds;

f+ε (t) =

12ε

∫ t+ε

t

f(s)ds; f−ε (t) =

12ε

∫ t

t−ε

f(s)ds

One considers quantum paths a la Feynman so that limt→t′ [X(t)−X(t′)]2/(t− t′)exists. This implies X(t) ∈ H1/2 where Hα means cεα ≤ |X(t)−X(t′)| ≤ Cεα andfrom Remark 1.2.1 for example this means dimHX[a, b] = 1/2. Next, thinking ofclassical Lagrangians L(x, v, t) = (1/2)mv2 + U(x, t), one defines an operator Qvia ((x, t, v) ∼ classical variables)

(2.29) Q(t) = t; Q(x(t)) = X(t); Q(v(t)) = V(t); Q

(df

dt

)= Q

(d

dt

)·Q(f)

where Q(d/dt) = d/dt if Q(f)(t) is differentiable and Q(d/dt) = ε/t whereε(x, h) > ε > 0 if Q(f)(t) is nondifferentiable. Note V(t) = Q(d/dt)[X(t)]so regularity of X determines the form of Q here and for Q(x) = X ∈ H1/2

one has V = εX/t. The scalar Euler-Lagrange (EL) equation associated toL(X(t),V(t), t) = Q(L(x(t), v(t), t) is

(2.30)ε

t

(∂L∂V (X(t),V(t), t)

)=

∂L∂X

(X(t),V(t), t)


Now given a classical v ∼ (1/m)(∂S/∂x) one gets V = (1/m)(∂S/∂X) and L =(∂S/∂t) with ψ(X, t) = exp[iS(X, t)/2mγ]. For L ∼ (1/2)mV2 + U then thequantum (EL) equation is m(εV/t) = (∂U/∂X) leading to(2.31)

2iγm

[−ψ2

X

ψ

(iγ +

aε(t)2

)+ ∂tψ +

aε(t)2

∂2ψ

∂X2

]= (U(X) + α(X))ψ + o(ε1/2)

where

(2.32) aε(t) =12[(∇ε

+X(t))2 − (∇ε−X(t))2]− i[(∇ε

+X(t))2 + (∇ε−X(t))2]

Then (2.32) is called the generalized SE and the nonlinear character of such equa-tions is discussed in [192, 223] for example. In [272] one then arrives at a con-ventional looking SE under the assumption aε = −2iγ, leading to

(2.33) γ2 ∂2ψ

∂X2+ iγ

∂ψ

∂t= [U(X, t) + α(X)]

ψ

2m+ o(ε1/2)

One can then always take α(X) = 0 and choosing γ = /2m one arrives atiψt + (2/2m)(∂2ψ/∂t2) = Uψ. However the requirement aε(t) = −2iγ seemsquite restrictive.

• Note here that the argument using a± is rigorous via [272]. aε = −i/mis permissible and in fact can have solutions of ∇ε

σX(t) = constant viaXc(t) = ±

√/2m)(t− c− (ε/2))+Pε(t) where Pε ∈ H1/2 is an arbitrary

periodic function.Referring back to Example 1.2.3 we have b±(t)(t) ∼ ±x(t) and V = (/2)(+x +−x)(t) with U = (1/2)(+x−−x)(t). The relation between U and the quantumpotential Q will formally still hold (cf. also [273] on nondifferentiable variationalprinciples) and one can rewrite this as

√ρU = (/m)∂

√ρ;√

ρQ = −(2/2m)∂2√ρalong with ∂(

√ρU) = −(2/)

√ρQ. If U is not differentiable one could also look at

√ρU = −(2/)

∫X

0

√ρQdX ′ + f(t) with f(t) possibly determinable via the term

(√

ρU)(0, t).

3. REMARKS ON FRACTAL SPACETIME

There have been a number of articles and books involving fractal methods inspacetime or fractal spacetime itself with impetus coming from quantum physicsand relativity. We refer here especially to [1, 186, 187, 225, 422, 675, 676,677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690]for background to this paper. Many related papers are omitted here and werefer in particular to the journal Chaos, Solitons, and Fractals CSF) for furtherinformation. For information on fractals and stochastic processes we refer forexample to [33, 83, 241, 242, 243, 345, 423, 555, 562, 592, 643, 625, 697,725, 748, 763, 810, 918, 942, 985]. We discuss here a few background ideasand constructions in order to indicate the ingredients for El Naschie’s Cantorianspacetime E∞, whose exact nature is elusive. Suitable references are given butthere are many more papers in the journal CSF by El Naschie (and others) basedon these fundamental ideas and these are either important in a revolutionarysense or a fascinating refined form of science fiction. In what appears at times

3. REMARKS ON FRACTAL SPACETIME 23

to be pure numerology one manages to (rather hastily) produce amazingly closenumerical approximations to virtually all the fundamental constants of physics(including string theory). The key concepts revolve around the famous goldenratio (

√5 − 1)/2 and a strange Cantorian space E∞ which we try to describe

below. It is very tempting to want all of these (heuristic) results to be true andthe approach seems close enough and universal enough to compel one to thinksomething very important must be involved. Moreover such scope and accuracycannot be ignored so we try to examine some of the constructions in a didacticmanner in order to possibly generate some understanding.

3.1. COMMENTS ON CANTOR SETS.

EXAMPLE 3.1. In the paper [643] one discusses random recursive con-structions leading to Cantor sets, etc. Associated with each such construction isa universal number α such that almost surely the random object has Hausdorffdimension α (we assume that ideas of Hausdorff and Minkowski-Bouligand (MB)or upper box dimension are known - cf. [83, 186, 345, 592]). One constructionof a Cantor set goes as follows. Choose x from [0, 1] according to the uniformdistribution and then choose y from [x, 1] according to the uniform distribution on[x, 1]. Set J0 = [0, x] and J1 = [y, 1] and recall the standard 1/3 construction forCantor sets. Continue this procedure by rescaling to each of the intervals alreadyobtained. With probability one one then obtains a Cantor set S0

c with Hausdorffdimension α = φ = (

√5− 1)/2 ∼ .618. Note that this is just a particular random

Cantor set; there are others with different Hausdorff dimensions (there seems tobe some - possibly harmless - confusion on this point in the El Naschie papers).However the golden ratio φ is a very interesting number whose importance rivalsthat of π or e. In particular (cf. [1]) φ is the hardest number to approximate byrational numbers and could be called the most irrational number. This is becauseits continued fraction represention involves all 1′s.

EXAMPLE 3.2. From [676] the Hausdorff (H) dimension of a traditionaltriadic Cantor set is d

(0)c = log(2)/log(3). To determine the equivalent to a

triadic Cantor set in 2 dimensions one looks for a set which is triadic Can-torian in all directions. The analogue of an area A = 1 × 1 is a quasi-areaAc = d

(0)c × d

(0)c and to normalize Ac one uses ρ2 = (A/Ac)2 = 1/(d(0)

c )2 (forn-dimensions ρn = 1/(d(0)

c )n−1). Then the nth Cantor like H dimension d(n)c

will have the form d(n)c = ρnd

(0)c = 1/(d(0)

c )n−1. Note also that the H dimen-sion of a Sierpinski gasket is d

(n+1)c /d

(n)c = 1/d

(0)c = log(3)/log(2) and in any

event the straight-forward interpretation of d(2)c = log(3)/log(2) is a scaling of

d(0)c = log(2)/log(3) proportional to the ratio of areas (A/Ac)2. One notes that

d(4)c = 1/(d(0)

c )3 = (log(3)/log(2))3 3.997 ∼ 4 so the 4-dimensional Cantor set isessentially “space filling”.

Another derivation goes as follows. Define probability quotients via Ω =dim(subset)/dim(set). For a triadic Cantor set in 1-D Ω(1) = d

(0)c /d

(1)c = d

(0)c (d(1)

c =1). To lift the Cantor set to n-dimensions look at the multiplicative probability

law Ω(n) = (Ω(1))n = (d(0)c )n. However since Ω(1) = d

(0)c /d

(n)c we get

(3.1) d(0)c /d(n)

c = (d(0)c )n ⇒ d(n)

c = 1/(d(0)c )n−1

Since Ω(n−1) is the probability of finding a Cantor point (Cantorian) one can thinkof the H dimension d

(n)c = 1/Ω(n−1) as a measure of ignorance. One notes here

also that for d(0)c = φ (the Cantor set S

(0)c of Example 3.1) one has d

(4)c = 1/φ3 =

4 + φ3 4.236 which is surely space filling.

Based on these ideas one proves in [676, 680, 682] a number of theorems andwe sketch some of this here. One picks a “backbone” Cantor set with H dimensiond(0)c (the choice of φ = d

(0)c will turn out to be optimal for many arguments). Then

one imagines a Cantorian spacetime E∞ built up of an infinite number of spacesof dimension d

(n)c (−∞ ≤ n < ∞). The exact form of embedding etc. here is not

specified so one imagines e.g. E∞ = ∪E(n) (with unions and intersections) in someamorphous sense. There are some connections of this to vonNeumann’s continuousgeometries indicated in [684]. In this connection we remark that only E(−∞) isthe completely empty set (E(−1) is not empty). First we note that φ2 + φ− 1 = 0leading to

(3.2)1 + φ = 1/φ, φ3 = (2 + φ)/φ, (1 + φ)/(1− φ) = 1/φ(1− φ) = 4 + φ3 = 1/φ3

(a very interesting number indeed). Then one asserts that

THEOREM 3.1. Let (Ω(1))n be a geometrical measure in n-dimensionalspace of a multiplicative point set process and Ω(1) be the Hausdorff dimensionof the backbone (generating) set d

(0)c . Then < d >= 1/d

(0)c (1 − d

(0)c ) (called

curiously an average Hausdorff dimension) will be exactly equal to the averagespace dimension ˜ < n >= (1 + d

(0)c )(1 − d

(0)c ) and equivalent to a 4-dimensional

Cantor set with H-dimension d(4)c = 1/(d(0)

c )3 if and only if d(0)c = φ.

To see this take Ω(n) = (Ω(1))n again and consider the total probability of theadditive set described by the Ω(n), namely Z0 =

∑∞0 (Ω(1))n = 1/(1 − Ω(1)). It

is conceptually easier here to regard this as a sum of weighted dimensions (sinced(n)c = 1/(d(0)

c )n−1) and consider wn = n(d(0)c )n. Then the expectation of n

becomes (note d(n)c ∼ 1/(d(0)

c )n−1 ∼ 1/Ω(n−1) so n(d(0)c )n−1 ∼ n/d

(n)c )

(3.3) E(n) =∑∞

1 n2(d(0)c )n−1∑∞

1 n(d(0)c )n−1

=˜< n >=1 + d

(0)c

1− d(0)c

Another average here is defined via (blackbody gamma distribution)

(3.4) < n >=

∫∞0

n2(Ω(1))ndn∫∞0

n(Ω(1))ndn=

−2log(Ω(1))

which corresponds to ˜ < n > after expanding the logarithm and omitting higherorder terms. However ˜ < n > seems to be the more valid calculation here. Simi-larly one defines (somewhat ambiguously) an expected value for d

(n)c via

(3.5) < d >=∑∞

1 n(d(0)c )n−1∑∞

1 (d(0)c )n

=1

d(0)c (1− d

(0)c )

This is contrived of course (and cannot represent E(d(n)c ) since one is computing

reciprocals∑

(n/d(n)c ) but we could think of computing an expected ignorance and

identifying this with the reciprocal of dimension. Thus the label < d > does notseem to represent an expected dimension but if we accept it as a symbol then ford(0)c = φ one has

(3.6) ˜< n >=1 + φ

1− φ=< d >=

1φ(1− φ)

= d(4)c = 4 + φ3 =

1φ3∼ 4.236

REMARK 1.3.1. We note that the normalized probability N = Ω(1)/Z0 =Ω(1)(1−Ω(1)) = 1/ < d > for any d

(0)c . Further if < d >= 4 = 1/d

(0)c (1− d

(0)c ) one

has d(0)c = 1/2 while˜< n >= 3 < 4 =< d >. One sees also that d

(0)c = 1/2 is the

minimum (where d < d > /d(d(0)c ) = 0).

REMARK 1.3.2. The results of Theorem 3.1 should really be phrased interms of E∞ (cf. [685]). thus (H ∼ Hausdorff dimension and T ∼ topologicaldimension)

(3.7) dimHE(n) = d(n)

c =1

(d(0)c )n−1

;

< d >=1

d(0)c (1− d

(0)c )

; ˜< dimT E∞ >=

1 + d(0)c

1− d(0)c

=˜< n >

In any event E∞ is formally infinite dimensional but effectively it is 4± dimen-sional with an infinite number of internal dimensions. We emphasize that E∞

appears to be constructed from a fixed backbone Cantor set with H dimension1/2 ≤ d

(0)c < 1; thus each such d

(0)c generates an E∞ space. Note that in [685] E∞

is looked upon as a transfinite discretum underpinning the continuum (whateverthat means).

REMARK 1.3.3. An interesting argument from [684] goes as follows.Thinking of d

(0)c as a geometrical probability one could say that the spatial (3-

dimensional) probability of finding a Cantorian “point” in E∞ must be given bythe intersection probability P = (d(0)

c )3 where 3 ∼ 3 topological spatial dimen-sion. P could then be regarded as a Hurst exponent (cf. [1, 715, 985]) and theHausdorff dimension of the fractal path of a Cantorian would be dpath = 1/H =1/P = 1/(d(0)

c )3. Given d(0)c = φ this means dpath = 4 + φ3 ∼ 4+ so a Cantorian

in 3-D would sweep out a 4-D world sheet; i.e. the time dimension is created bythe Cantorian space E∞ (! - ?). Conjecturing further (wildly) one could say thatperhaps space (and gravity) is created by the fractality of time. This is a typical


form of conjecture to be found in the El Naschie papers - extremely thought pro-voking but ultimately heuristic. Regarding the Hurst exponent one recalls thatfor Feynmann trajectories in 1 + 1 dimensions dpath = 1/H = 1/d

(0)c = d

(2)c . Thus

we are concerned with relating the two determinations of dpath (among other mat-ters). Note that path dimension is often thought of as a fractal dimension (M-Bor box dimension), which is not necessarily the same as the Hausdorff dimension.However in [29] one shows that quantum mechanical free motion produces fractalpaths of Hausdorff dimension 2 (cf. also [583]).

REMARK 1.3.4. Following [226] let S(0)c correspond to the set with di-

mension d(0)c = φ. Then the complementary dimension is d

(0)c = 1 − φ = φ2.

The path dimension is given as in Remark 1.3.3 by dpath = d(2)c = 1/φ = 1 + φ

and dpath = d(2)c = 1/(1 − φ) = 1/φ2 = (1 + φ)2. Following El Naschie for an

equivalence between unions and intersections in a given space one requires (in thepresent situation) that

(3.8)

dcrit = d(2)c + d

(2)c = 1

φ + 1φ2 = φ(1+φ)

φ3 = 1φ3 = 1

φ ·1

φ2 = d(2)c · d(2)

c = 4 + φ3

where dcrit = 4 + φ3 = d(4)c ∼ 4.236. Thus the critical dimension coincides with

the Hausdorff dimension of S(4)c which is embedded densely into a smooth space

of topological dimension 4. On the other hand the backbone set of dimensiond(0)c = φ is embedded densely into a set of topological dimension zero (a point).

Thus one thinks in general of d(n)c as the H dimension of a Cantor set of dimension

φ embedded into a smooth space of integer topological dimension n.

REMARK 1.3.5. In [226] it is also shown that realization of the spaces E(n)

comprising E∞ can be expressed via the fractal sprays of Lapidus-van Frankenhuy-sen (cf. [592]). Thus we refer to [592] for graphics and details and simply sketchsome ideas here (with apologies to M. Lapidus). A fractal string is a boundedopen subset of R which is a disjoint union of an infinite number of open inter-vals L = 1, 2, · · · . The geometric zeta function of L is ζL(s) =

∑∞1 −s

j . Oneassumes a suitable meromorphic extension of ζL and the complex dimensions ofL are defined as the poles of this meromorphic extension. The spectrum of L

is the sequence of frequencies f = k · −1j (k = 1, 2, · · · ) and the spectral zeta

function of L is defined as ζν(s) =∑

f f−s where in fact ζν(s) = ζL(s)ζ(s) (withζ(s) the classical Riemann zeta function). Fractal sprays are higher dimensionalgeneralizations of fractal strings. As an example consider the spray Ω obtainedby scaling an open square B of size 1 by the lengths of the standard triadic Can-tor string CS. Thus Ω consists of one open square of size 1/3, 2 open squares ofsize 1/9, 4 open squares of size 1/27, etc. (see [592] for pictures and explana-tions). Then the spectral zeta function for the Dirichlet Laplacian on the squareis ζB(s) =

∑∞n1,n2=1(n

21 + n2

2)s/2 and the spectral zeta function of the spray is

ζν(s) = ζCS(s) · ζB(s). Now E∞ is composed of an infinite hierarchy of sets E(j)

with dimension (1+φ)j−1 = 1/φj−1 (j = 0,±1,±2, · · · ) and these sets correspond


to a special case of boundaries ∂Ω for fractal sprays Ω whose scaling ratios aresuitable binary powers of 2−φj−1

. Indeed for n = 2 the spectral zeta function ofthe fractal golden spray indicated above is ζν(s) = (1/(1 − 2 · 2sφ)ζB(s). Thepoles of ζB(s) do not coincide with the zeros of the denominator 1 − 2 · 2−sφ sothe (complex) dimensions of the spray correspond to those of the boundary ∂Ωof Ω. One finds that the real part s of the complex dimensions coincides withdimE(2) = 1+φ = 1/φ2 and one identifies then ∂Ω with E(2). The procedure gen-eralizes to higher dimensions (with some stipulations) and for dimension n thereresults s = 1/φn−1 = dimE(n). This produces a physical model of the Cantorianfractal space from the boundaries of fractal sprays (see [226] for further details and[592] for precision). Other (putative) geometric realizations of E∞ are indicatedin [688] in terms of wild topologies, etc.

3.2. COMMENTS ON HYDRODYNAMICS. We sketch first some ma-

Thus let ψ be the wave function of a test particle of mass m0 in a force fieldU(r, t) determined via i∂tψ = Uψ − (2/2m)∇2ψ where ∇2 = ∆. One writesψ(r, t) = R(r, t)exp(iS(r, t)) with v = (/2m)∇S and ρ = R · R (one assumesρ = 0 for physical meaning). Thus the field equations of QM in the hydrodynamicpicture are

(3.9) dt(m0ρv) = ∂t(m0ρv) +∇(m0ρv) = −ρ∇(U + Q); ∂tρ +∇ · (ρv) = 0

where Q = −(2/2m0)(∆√

ρ/√

ρ) is the quantum potential (or interior potential).Now because of the nondifferentiability of spacetime an infinity of geodesics willexist between any couple of points A and B. The ensemble will define the proba-bility amplitude (this is a nice assumption but geodesics should be defined here).At each intermediate point C one can consider the family of incoming (backward)and outgoing (forward) geodesics and define average velocities b+(C) and b−(C)on these families. These will be different in general and following Nottale thisdoubling of the velocity vector is at the origin of the complex nature of QM. Eventhough Nottale reformulates Nelson’s stochastic QM the former’s interpretation isprofoundly different. While Nelson (cf. [698]) assumes an underlying Brownianmotion of unknown origin which acts on particles in Minkowskian spacetime, andthen introduces nondifferentiability as a byproduct of this hypothesis, Nottale as-sumes as a fundamental and universal principle that spacetime itself is no longerMinkowskian nor differentiable. An interesting comment here from [15] is thatwith Nelson’s Browian motion hypothesis, nondifferentiability is but an approx-imation which expected to break down at the scale of the underlying collisions,where a new physics should be introduced, while Nottale’s hypothesis of nondiffer-entiability is essential and should hold down to the smallest possible length scales.Following Nelson one defines now the mean forward and backward derivatives

(3.10)d±dt

y(t) = lim∆t→0±

⟨y(t + ∆t)− y(t)

∆t

⟩This gives forward and backward mean velocities (d+/dt)x(t) = b+ and (d−/dt)x(t) =b− for a position vector x. Now in Nelson’s stochastic mechanics one writes twosystems of equations for the forward and backward processes and combines them in

terial from [15] (see also [294, 715, 718, 720] and Sections 1-2 for background).

the end in a complex equation, Nottale works from the beginning with a complexderivative operator

(3.11)δ

dt=

(d+ + d−)− i(d+ − d−)2dt

leading to V = (δ/dt)x(t) = v − iu = (1/2)(b+ + b−) − (i/2)(b+ − b−). Onedefines also (dv/dt) = (1/2)(d+ +d−)/dt and (du/dt) = (1/2)(d+−d−)/dt so thatdvx/dt = v and dux/dt = u. Here v generalizes the classical velocity while u is anew quantity arising from nondifferentiability. This leads to a stochastic processsatisfying (respectively for the forward (dt > 0) and backward (dt < 0) processes)dx(t) = b+[x(t)] + dξ+(t) = b−[x(t)] + dξ−(t). The dξ(t) terms can be seen asfractal functions and they amount to a Wiener process when the fractal dimensionD = 2. Then the dξ(t) are Gaussian with mean zero, mutually independent, andsatisfy < dξ±idξ±j >= ±2Dδijdt where D is a diffusion coefficient determinedas D = /2m0 when τ0 = /(m0c

2) (deBroglie time scale in the rest frame (cf.[15]). This allows one to give a general expression for the complex time derivative,namely

(3.12) df =∂f

∂t+∇f · dx +

12

∂2f

∂xi∂xjdxidxj

Next compute the forward and backward derivatives of f ; then one will arrive at< dxidxj >→< dξ±idξ±j > so the last term in (3.12) amounts to a Laplacianand one obtains (d±f/dt) = [∂t + b± · ∇ ± D∆]f which is an important result.Thus assume the fractal dimension is not 2 in which case there is no longer acancellation of the scale dependent terms in (3.12) and instead of D∆f one wouldobtain an explicitly scale dependent behavior proportional to δt(2/D)−1∆f . Inother words the value D = 2 for the fractal dimension implies that the scalesymmetry becomes hidden in the operator formalism. One obtains the complextime derivative operator in the form (δ/dt) = ∂t + V · ∇ − iD∆ (V as above).Nottale’s prescription is then to replace d/dt by δ/dt. In this spirit one can writenow ψ = exp(i(S/2m0D)) so that V = −2iD∇(log(ψ)) and then the generalizedNewton equation −∇U = m0(δ/dt)V reduces to the SE (L = (1/2)mv2 − U).

Now assume the velocity field from the hydrodynamic model agrees with thereal part v of the complex velocity V and equate the wave functions from the twomodels ψ = exp(iS/2m0D) and ψ = Rexp(iS) with m = m0; one obtains forS = s + iσ the formulas s = 2m0DS, D = (/2m0), and σ = −m0Dlog(ρ). Usingthe definition V = (1/m0)∇S = (1/m0)∇s + (i/m0)∇σ = v − iu (which resultsfrom the above equations) we get

(3.13) v = (1/m0)∇s = 2D∇S; u = −(1/m0)∇σ = D∇log(ρ)

Note that the imaginary part of the complex velocity coincides with Nottale. Di-viding the time dependent SE iψt = Uψ − (2/2m0)∆ψ by 2m0 and takingthe gradient gives ∇U/m0 = 2D∇[i∂tlog(ψ) + D(∆ψ/ψ)] where /2m0 has beenreplaced by D. Then consider the identities

(3.14) ∆∇ = ∇∆; (∇f · ∇)(∇f) = (1/2)∇(∇f)2;∆f

f= ∆log(f) + (∇log(f))2


Then the second term in the right of the equation for∇U/m0 becomes∇(∆ψ/ψ) =∆(∇log(ψ)) + 2(∇log(ψ) · ∇)(∇log(ψ)) so we obtain

(3.15) ∇U = 2iDm0[∂t∇log(ψ)− iD∆(∇log(ψ)− 2iD(∇log(ψ) · ∇)(∇log(ψ))]

One can show that this is nothing but the generalized Newton equation −∇U =m0(δ/dt)V . Now replacing the complex velocity V = −2iD∇log(ψ) and takinginto account the form of V, we get

(3.16) −∇U = m0∂t(v − iD∇log(ρ) + [i(v − iD∇log(ρ) · ∇](v − iD∇log(ρ))−

−iD∆(v − iD∇log(ρ))Equation (3.16) is a complex differential equation and reduces to

(3.17) m0[∂tv + (v · ∇)v] = −∇(

U − 2m0D2 ∆√

ρ√

ρ

); ∇

1ρ

[∂tρ +∇ · (ρv)]

The last equation in (3.17) reduces to the continuity equation up to a phase factorα(t) which can be set equal to zero (note again that ρ = 0 is posited). Thus(3.17) is nothing but the fundamental equations (3.9) of the hydrodynamic model.Further combining the imaginary part of the complex velocity with the quantumpotential, and using (3.14), one gets Q = −m0D∇·u− (1/2)m0u

2 (as indicated inRemark 1.2.2). Since u arises from nondifferentiability according to our nondiffer-entiable space model of QM it follows that the quantum potential comes from thenondifferentiability of the quantum spacetime (note that the x derivatives shouldbe clarified and E∞ has not been utilized).

Putting U = 0 in the first equation of (3.17), multiplying by ρ, and taking thesecond equation into account yields

(3.18) ∂t(m0ρνk) +∂

∂xi(m0ρνiνk) = −ρ

∂

xk

[2m0D2 1

√ρ

∂

∂xi

∂

∂xi(√

ρ)]

(here νk ∼ vk seems indicated). Now set Πik = m0ρνiνk − σik along with σik =m0ρD2(∂/∂xi)(∂/∂xk)(log(ρ)). Then (3.18) takes the simple form

(3.19) ∂t(m0ρνk) = −∂Πik/∂xi

The analogy with classical fluid mechanics works well if one introduces the kine-matic µ = D/2 and dynamic η = (1/2)m0Dρ viscosities. Then Πik defines the mo-mentum flux density tensor and σik the internal stress tensor σik = η[(∂ui/∂xk)+(∂uk/∂xi)]. One can see that the internal stress tensor is build up using thequantum potential while the equations (3.18) or (3.19) are nothing but systems ofNavier-Stokes type for the motion where the quantum potential plays the role ofan internal stress tensor. In other words the nondifferentiability of the quantumspacetime manifests itself like an internal stress tensor. For clarity in understand-ing (3.19) we put this in one dimensional form so (3.18) becomes

(3.20) ∂t(m0ρv) + ∂x(m0ρv2) = −ρ∂

(2m0D2 1

√ρ∂2√ρ

)= ρ∂Q

and Π = m0ρv2−σ with σ = m0ρD2∂2log(ρ) which agrees with standard formulas.Now note ∂

√ρ = (1/2)ρ−1/2ρ′ and ∂2√ρ = (1/2)[−(1/2)ρ−3/2(ρ′)2+ρ−1/2ρ′′] with

∂2log(ρ) = ∂(ρ′/ρ) = (ρ′′/ρ)− (ρ′/ρ)2 while(3.21)

−ρ∂

[2m0D2 1

√ρ

(∂2√ρ

)]= −2m0D2ρ∂

[1

2√

ρ

(−1

2ρ−3/2(ρ′)2 + ρ−1/2ρ′′

)]=

= −2m0D2ρ∂

[ρ′′

2ρ− 1

4

((ρ′

ρ

)2]

= −m0D2ρ∂

[ρ′′

ρ− 1

2

(ρ′

ρ

)2]

One wants to show then that (3.19) holds or equivalently −∂σ = (3.21). However(3.22)

−∂σ = −∂[m0ρD2∂2log(ρ)] = −m0D2

[ρ′(

ρ′′

ρ−(

ρ′

ρ

)2)

+ ρ∂

(ρ′′

ρ− (ρ′)2

ρ

)]so we want (3.22) = (3.21) which is easily verified.

4. REMARKS ON FRACTAL CALCULUS

We sketch first (in summary form) from [748] where a calculus based on fractalsubsets of the real line is formulated. A local calculus based on renormalizingfractional derivatives a la [562] is subsumed and embellished. Consider first theconcept of content or α-mass for a (generally fractal) subset F ⊂ [a, b] (in whatfollows 0 < α ≤ 1). Then define the flag function for a set F and a closed intervalI as θ(F, I) = 1 (F ∩ I = ∅ and otherwise θ = 0. Then a subdivision P[a,b] ∼ P of[a, b] (a < b) is a finite set of points a = x0, x1, · · · , xn = b with xi < xi+1. If Qis any subdivision with P ⊂ Q it is called a refinement and if a = b the set a isthe only subdivision. Define then

(4.1) σα[F, p] =n−1∑

0

(xi+1 − xi)α

Γ(α + 1)θ(F, [xi, xi+1])

For a = b one defines σα[F, P ] = 0. Next given δ > 0 and a ≤ b the coarse grainedmass γα

δ (F, a, b) of F ∩ [a, b] is given via

(4.2) γαδ (F, a, b) = inf|P |≤δσ

α[F, P ] (|P | = max0≤i≤n−1(xi+1 − xi))

where the infimum is over P such that |P | ≤ δ. Various more or less straightforwardproperties are:

• For a ≤ b and δ1 < δ2 one has γαδ1

(F, a, b) ≥ γαδ2

(F, a, b).• For δ > 0 and a < b < c one has γα

δ (F, a, b) ≤ γαδ (F, a, c) and γα

δ (F, b, c) ≤γα

δ (F, a, c).• γα

δ is continuous in b and a.Now define the mass function γα(F, a, b) via γα(F, a, b) = limδ→0γ

αδ (F, a, b). The

following results are proved(1) If F ∩ (a, b) = ∅ then γα(F, a, b) = 0.(2) Let a < b < c and γα(F, a, c) < ∞. Then γα(F, a, c) = γα(F, a, b) +

γα(F, b, c). Hence γα(F, a, b) is increasing in b and decreasing in a.

4. REMARKS ON FRACTAL CALCULUS 31

(3) Let a < b and γα(F, a, b) = 0 be finite. If 0 < y < γα(F, a, b) then thereexists c, a < c < b such that γα(F, a, c) = y. Further if γα(F, a, b) isfinite then γα(F, a, x) is continuous for x ∈ (a, b).

(4) For F ⊂ R and λ ∈ R let F + λ = x + λ; x ∈ F. Then γα(F + λ, a +λ, b + λ) = γα(F, a, b) and γα(λF, λa, λb) = λαγα(F, a, b).

Now for a0 an arbitrary fixed real number one defines the integral staircase functionof order α for F is

(4.3) SαF (x) =

γα(F, a0, x) x ≥ a0

−γα(F, x, a0) otherwise

The following properties of SF are restatements of properties for γα. thus• Sα

F (x) is increasing in x.• If F ∩ (x, y) = 0 then Sα

F is constant in [x, y].• Sα

F (y)− SαF (x) = γα(F, x, y).

• SαF is continuous on (a, b).

Now one considers the sets F for which the mass function γα(F, a, b) gives themost useful information. Indeed one can use the mass function to define a fractaldimension. If 0 < α < β ≤ 1 one writes(4.4)

σβ [F, P ] ≤ |P |β−ασα[F, P ]Γ(α + 1)Γ(β + 1)

; γβδ (F, a, b) ≤ δβ−αγα

δ (F, a, b)Γ(α + 1)Γ(β + 1)

Thus in the limit δ → 0 one gets γβ(F, a, b) = 0 provided γα(F, a, b) < ∞ andα < β. It follows that γα(F, a, b) is infinite up to a certain value α0 and then jumpsdown to zero for α > α0 (if α0 < 1). This number is called the γ-dimension ofF; γα0(F, a, b) may itself be zero, finite, or infinite. To make the definition preciseone says that the γ-dimension of F ∩ [a, b], denoted by dimγ(F ∩ [a, b]), is

(4.5) dimγ(F ∩ [a, b]) =

infα; γα(F, a, b) = 0supα; γα(F, a, b) = ∞

One shows that dimH(F ∩ [a, b]) ≤ dimγ(F ∩ [a, b]) where dimH denotes Hausdorffdimension. Further dimγ(F ∩ [a, b]) ≤ dimB(F ∩ [a, b]) where dimB is the box di-mension. Some further analysis shows that for F ⊂ R compact dimγF = dimHF .

Next one notes that the correspondence F → SαF is many to one (examples

from Cantor sets) and one calls the sets giving rise to the same staircase function“staircasewise congruent”. The equivalence class of congruent sets containing F isdenoted by EF ; thus if G ∈ EF it follows that Sα

G = SαF and Eα

G = EαF . One says

that a point x is a point of change of f if f is not constant over any open interval(c, d) containing x. The set of all points of change of f is denoted by Sch(f). Inparticular if G ∈ Eα

F then SαG(x) = Sα

F (x) so Sch(SαG) = Sch(Sα

F ). Thus if F ⊂ Ris such that Sα

F (x) is finite for all x (α = dimγF ) then H = Sch(SαF ) ∈ Eα

F . Thistakes some proving which we omit (cf. [748]). As a consequence let F ⊂ R besuch that Sα

F (x) is finite for all x ∈ R (α = dimγF ). Then the set H = Sch(SαF ) is

perfect (i.e. H is closed and every point is a limit point). Hence given SαF (x) finite

for all x (α = dimγF ) one calls Sch(SαF ) the α-perfect representative of Eα

F and oneproves that it is the minimal closed set in Eα

F . Indeed an α-perfect set in EαF is the

intersection of all closed sets G in EαF . One can also say that if F ⊂ R is α-perfect

and x ∈ F then for y < x < z either SαF (y) < Sα

F (x) or SαF (x) < Sα

F (z) (or both).Thus for an α-perfect set it is assured that the values of Sα

F (y) must be differentfrom Sα

F (x) at all points y on at least one side of x. As an example one shows thatthe middle third Cantor set C = E1/3 is α-perfect for α = log(2)/log(3) = dH(C)so C = Sch(Sα

C).

Now look at F with the induced topology from R and consider the idea ofF-continuity.

DEFINITION 4.1. Let F ⊂ R and f : R → R with x ∈ F . A number is said to be the limit of f through the points of F, or simply F-limit, as y → x ifgiven ε > 0 there exists δ > 0 such that y ∈ F and |y − x| < δ ⇒ |f(y) − | < ε.In such a case one writes = F − limity→xf(y). A function f is F-continuous atx ∈ F if f(x) = F − limity→xf(y) and uniformly F-continuuous on E ⊂ F if forε > 0 there exists δ > 0 such that x ∈ F, y ∈ E and |y−x| < δ ⇒ |f(y)−f(x)| < ε.One sees that if f is F-continuous on a compact set E ⊂ F then it is uniformlyF-continuous on E.

DEFINITION 4.2. The class of functions f : R → R which are boundedon F is denoted by B(F ). Define for f ∈ B(F ) and I a closed interval

(4.6) M [f, F, I] =

supx∈F∩If(x) F ∩ I = ∅0 otherwise

m[f, F, I] =

infx∈F∩If(x) F ∩ I = ∅0 otherwise

DEFINITION 4.3. Let SαF (x) be finite for x ∈ [a, b] and P be a subdivi-

sion with points x0, · · · , xn. The upper Fα and lower Fα sums over P are givenrespectively by

(4.7) Uα[f, F, P ] =n−1∑

0

M [f, F, [xi, xi+1]](SαF (xi+1)− Sα

F (xi));

Lα[f, F, P ] =n−1∑

0

m[f, F, [xi, xi+1]](SαF (xi+1)− Sα

F (xi))

This is sort of like Riemann-Stieltjes integration and in fact one shows that if Qis a refinement of P then Uα[f, F,Q] ≤ Uα[f, F, P ] and Lα[f, F,Q] ≥ Lα[f, F, P ].Further Uα[f, F, P ] ≥ Lα[f, F,Q] for any subdivisions of [a, b] and this leads tothe idea of F-integrability. Thus assume Sα

F is finite on [a, b] and for f ∈ B(F )one defines lower and upper Fα-integrals via

(4.8)∫ b

a

f(x)dαF x = supP Lα[f, F, P ];

∫ b

a

f(x)dαF x = infP Uα[f, F, P ]

One then says that f is Fα-integrable if (D15)∫ b

af(x)dα

F x =∫ b

af(x)dα

F x =∫ b

af(x)dα

F x.

4. REMARKS ON FRACTAL CALCULUS 33

One shows then(1) f ∈ B(F ) is Fα-integrable on [a, b] if and only if for any ε > 0 there is a

subdivision P of [a, b] such that Uα[f, F, P ] < Lα[f, F, P ] + ε.(2) Let F ∩[a, b] be compact with Sα

F finite on [a, b]. Let f ∈ B(F ) and a < b;then if f is F-continuous on F ∩ [a, b] it follows that f is Fα-integrableon [a, b].

(3) Let a < b and f be Fα-integrable on [a, b] with c ∈ (a, b). Then f is Fα-integrable on [a, c] and [c, b] with

∫ b

af(x)dα

F x =∫ c

af(x)dα

F x+∫ b

cf(x)dα

F x.(4) If f is Fα-integrable then

∫ b

aλf(x)dα

F x = λ∫ b

af(x)dα

F x and, for g alsoFα-integrable,

∫ b

a(f(x) + g(x))dα

F x =∫ b

af(x)dα

F x +∫ b

ag(x)dα

F x.(5) If f, g are Fα-integrable and f(x) ≥ g(x) for x ∈ F∩[a, b] then

∫ b

af(x)dα

F x ≥∫ b

ag(x)dα

F x.

One specifies also∫ a

bf(x)dα

F x = −∫ b

af(x)dα

F x and it is easily shown that if χF (x)is the characteristic function of F then

∫ b

aχF (x)dα

F x = SαF (b) − Sα

F (a). Now fordifferentiation one writes

(4.9) DαF f(x) =

F − limy→x

f(y)−f(x)Sα

F (y)−SαF (x) x ∈ F

0 otherwise

if the limit exists. One shows then(1) If Dα

F f(x) exists for all x ∈ (a, b) then f(x) is F-continuous in (a, b).(2) With obvious hypotheses Dα

F (λf(x)) = λDαF f(x) and Dα

F (f + g)(x) =Dα

F f(x) +DαF g(x). Further if f is constant then Dα

F f = 0.(3) Dα

F (SαF (x)) = χF (x).

(4) (Rolle’s theorem) Let f : R → R be continuous with Sch(f) ⊂ Fwhere F is α-perfect and assume Dα

F f(x) is defined for all x ∈ [a, b] withf(a) = f(b) = 0. Then there is a point c ∈ F∩[a, b] such that Dα

F f(c) ≥ 0and a point d ∈ F ∩ [a, b] where Dα

F f(d) ≤ 0.

EXAMPLE 4.1. This is the best that can be done with Rolle’s theorem sincefor C the Cantor set E1/3 take f(x) = Sα

C(x) for 0 ≤ x ≤ 1/2 and f(x) = 1−SαC(x)

for 1/2 < x ≤ 1. This function is continuous with f(0) = f(1) = 0 and the setof change (Sch(f)) is C. The Cα-derivative is given by Dα

Cf(x) = χC(x) for0 ≤ x ≤ 1/2 and by −χC(x) for 1/2 < x ≤ 1. Thus x ∈ C which impliesDα

Cf(x) = ±1 = 0.

As a corollary one has the following result: Let f be continuous with Sch(f) ⊂ Fwhere F is α-perfect; assume Dα

F f(s) exists at all points of [a, b] and that SαF (b) =

SαF (a). Then there are points c, d ∈ F such that

(4.10) DαF f(c) ≥ f(b)− f(a)

SαF (b)− Sα

F (b); Dα

F f(d) ≤ f(b)− f(a)Sα

F (b)− SαF (a)

Similarly if f is continuous with Sch(f) ⊂ F and DαF f(x) = 0 ∀x ∈ [a, b] then

f(x) is constant on [a, b]. There are also other fundamental theorems as follows(1) (Leibniz rule) If u, v : R → R are Fα-differentiable then Dα

F (uv)(x) =(Dα

F u(x))v(x) + u(x)DαF v(x).


(2) Let F ⊂ R be α-perfect. If f ∈ B(F ) is F-continuous on F ∩ [a, b] withg(x) =

∫ x

af(y)dα

F y for all x ∈ [a, b] then DαF g(x) = f(x)χF (x).

(3) Let f : R → R be continuous and Fα-differentiable with Sch(f) con-tained in an α-perfect set F; let also h : R → R be F-continuous suchthat h(x)χF (x) = Dα

F f(x). Then∫ b

ah(x)dα

F x = f(b)− f(a).(4) (Integration by parts) Assume: (i) u is continuous on [a, b] and Sch(u) ⊂

F . (ii) DαF u(x) exists and is F-continuous on [a, b]. (iii) v is F-continuous

on [a, b]. Then

(4.11)∫ b

a

uvdαF x =

[u(x)

∫ x

a

v(x′)dαF x′]

∣∣∣∣ba

−∫ b

a

DαF u(x)

∫ x

a

v(x′)dαF x′dα

F x

Some examples are given relative to applications and we mention e.g.

EXAMPLE 4.2. Following [562] one has a local fractal diffusion equation

(4.12) DαF,t(W (x, t)) =

χF (t)2

∂2

∂x2W (x, t)

with solution

(4.13) W (x, t) =1

(2πSαF (t))1/2

exp

(−x2

2SαF (t)

)The appendix to [748] also gives some formulas for repeated integration and

differentiation. For example it is shown that

(4.14) (DαF )2(Sα

F (x))2 = 2χF (x);∫ x′

a

(SαF (x))ndα

F x =1

n + 1(Sα

F (x′))n+1

We refer to [562, 748] for other interesting material.

5. A BOHMIAN APPROACH TO QUANTUM FRACTALS

The powerful exact uncertainty method of Hall and Reginatto for passingfrom classical to quantum mechanics has been further embellished and deepenedin recent years (see e.g. 444, 445, 446, 447, 448,449,In [445] one finds an apparent incompleteness in the traditional trajectory basedBohmian mechanics when dealing with a quantum particle in a box. It turns outthat there is no suitable HJ equation for describing the motion which in fact hasa fractal character. After reviewing the material on scale relativity in Section1.2 for example it is not surprising to encounter such situations and in [844] theBohmian point of view is reinstated for fractal trajectories. One should also remarkin passing that there is much material available on weak or distribution solutionsof HJ type equations and some of this should come into play here (cf. [211]). Themain issue here however is that in order to treat wave functions displaying fractalfeatures (quantum fractals) one needs to enlarge the picture via limiting processes.One derives the quantum trajectories by means of limiting procedures that involvethe expansion of the wave function in a series of eigenvectors of the Hamiltonian.

450, 749, 805, 806, 807, 844, 845] and Sections 1.1, 3. 1, and 4.7.[186, 187, 189, 203, 396,

5. A BOHMIAN APPROACH TO QUANTUM FRACTALS 35

Consider first the quantum analogue of the Weierstrass function

(5.1) W (x) =∞∑0

brSin(arx); a > 1 > b > 0; ab ≥ 1

Then in the problem of a particle in a 1-dimensional box of length L (with 0 <x < L) one can construct wave functions of the form

(5.2) Φt(x;R) = A

R∑r=0

nr(s−2)Sin(pn,rx/)e−iEn,rt/

with 2 > s > 0 and n ≥ 2. Here pn,r = nrπ/L is the quantized momentum (withinteger quantum number given by n′ = nr). En,r = p2

n,r/2m is the eigenenergyand a is a normalization constant. This wave function, which is a solution ofthe time dependent SE, is continuous and differentiable everywhere. However thewave function resulting in the limit, namely Φt(x) = limR→∞Φt(x;R) is a fractalobject in both space and time (cf. [626]). This method for generating quantumfractals basically involves (given s) choosing a quantum number, say n, and thenconsidering the series that contains its powers n′ = nr. There is also anotherrelated method (cf. [142]) of generating quantum fractals based on the presenceof discontinuities in the wave function. The emergence of fractal features arisesfrom the perturbations that such discontinuities cause in the wave function duringpropagation. This generating process can be easily understood by considering awave function initially uniform along a certain interval = x2− x1 ≤ L inside thebox

(5.3) Ψ0(x) = 1√

x1 < x < x2

0 otherwise

The Fourier decomposition of this wave function is

(5.4) Ψ0(x) =2

π√

∞∑1

1n

[Cos(pnx1/)− Cos(pnx2/)]Sin(pnx/)

whose time evolved form is

(5.5) Ψt(x) =2

π√

∞∑1

1n

[Cos(pnx1/)− Cos(pnx2/)]Sin(pnx/)e−iEnt/

It is equivalent to consider r = R = 1 in (5.2) and sum over n from 1 to N; thequantum fractal is then obtained in the limit N → ∞. This equivalence is basedon the fact that the Fourier decomposition of Ψ0 gives precisely its expansion interms of the eigenvectors of the Hamiltonian in the problem of a particle in a box(this is not a general situation).

EXAMPLE 5.1. The fractality of wave functions like Φt(x) or Ψt(x) canbe analytically estimated (cf. [142]) by taking advantage of a result for Fourierseries. Thus given an arbitrary function f(x) =

∑N1 anexp(−inx) its real and

imaginary pars are fractals (and also |f(x)|2) with dimension Df = (5−β)/2 if itspower spectrum has the asymptotic form |an|2 ∼ n−β for N →∞ with 1 < β ≤ 3.Alternatively the fractality of f(x) can also be calculated by measuring the length

L of its real and imaginary parts (or |f(x)|2) as a function of the number of termsN considered in the generating sries. Asymptotically the relation bewteen L andN can be expressed as L(N) ∝ NDf−1 which diverges if f(x) is a fractal object.One notes that to increase the number of terms contributing to f(x) is analogousto measuring its length with more precision.

It is known that for quantum fractals the corresponding expected value of theenergy < H > becomes infinite. This is related to the fact that the familiar formof the SE

(5.6) i∂tΨt(x) = HΨt(x)

does not hold in general (cf. [445, 1003]). In this case neither the left side of(5.6) nor the right side belong to the Hilbert space; however the identity

(5.7) [H − i∂t]Ψt(x) = 0

still remains valid. In this situation one says that Ψt(x) is a weak solution of theSE (note weak solutions have many meanings and have been extensively studiedin PDE - cf. [211]).

The formal basis of Bohmian mechanics (BM) is usually established via

(5.8) Ψt(x) = ρ1/2t (x)eiSt(x)/;

∂ρt

∂t+∇ ·

(ρt∇St

m

)= 0;

∂St

∂t+

(∇St)2

2m+ V + Qt = 0; Qt = − 2

2m

∇2ρ1/2t

ρ1/2t

One postulates also the trajectory velocity as

(5.9) x =∇St

m=

m[Ψ−1

t ∇Ψt]

Now Qt in (5.8) is well defined provided that the quantum state is also well defined(i.e. continuous and differentiable). However this is not the case for quantumfractals and the theory seems incomplete; the solution is to take into account thedecomposition of the quantum fractal in terms of differentiable eigenvectors andredefining Qt in (5.8). Thus any wave function Ψt is expressible as

(5.10) Ψt(x : N) =N∑1

cnξn(x)e−iEnt/

in the limit N → ∞ (cf. Φt above and (5.5)) where the ξn(x) are eigenvectorswith eigenvalues En of the corresponding Hamiltonian. One can then define thequantum trajectories evolving under the guidance of this wave as

(5.11) xt = limN→∞xN (t); xN =

m[Ψ−1

t (x;N)∂Ψt(x;N)

∂x

]Note the calculation of trajectories is not based on St, which has no trivial de-composition in a series of nice functions, but this kind of velocity formulation iscommon in e.g. [324, 325, 326, 327, 328, 329, 415, 416, 418, 419, 420] whereone modern version of BM is being developed.

5. A BOHMIAN APPROACH TO QUANTUM FRACTALS 37

EXAMPLE 5.2. A numerical example is given in [844] and we only mentiona few features here. Thus one considers a highly delocalized particle in a box withwave function (5.4) and x1 = 0 with x2 = L. Then (5.5) becomes

(5.12) Ψt(x) =4

π√

Le−iE1t/

∑n odd

1n

Sin(pnx/)eiωn,1t

where λn,1 = (En−E1)/ (in the numerical calculations one uses L = m = = 1).Here the probability density ρt is periodic in time but the wave function is notperiodic (this does not affect (5.11)). Various features are observed (e.g. Cantorset structures, Gibbs phenomena, etc.) and graphs are displayed - we omit anyfurther discussion here.

In summary, although the SE is not satisfied by quantum fractals as a whole, itis when one considers its decomposition in terms of the eigenvectors of the Hamil-tonian. The contributing eigenvectors are continuous and differentiable and anywave function (regular or not) admits a decomposition in terms of eigenvectors.Correspondingly the Bohmian equation of motion must be reformulated in termsof such decompositions via (5.11) and this can be regarded also as a generalizationof (5.8). We mention in passing that from time to time there are papers claimingcontradictions between BM and QM and we refer here to [436, 629, 712] for somerefutations.

REMARK 5.3. Let us mention here a suggestion of ’t Hooft [475] aboutestablishing the physical link between classical and quantum mechanics by em-ploying the underlying equations of classical mechanics and including into them aspecially chosen dissipative function. The wave like QM turns out to follow fromthe particle like classical mechanics due to embedding in the latter a dissipation“device” responsible for loss of information. Thus the initial precise informationabout the classical trajectory is lost in QM due to the “dissipative spread” of thetrajectory and its transformation into a fuzzy object such as the fractal Hausdorffpath of dimension 2 in a simple case of a spinless particle. Some rough calculationsin this direction appear in [426]. and we refer also to [122, 427, 749].

CHAPTER 2

DEBROGLIE-BOHM IN VARIOUS CONTEXTS

The quantum potential arises in various forms, some of which were discussedin Sections 1.1 and 1.2. We return to this now in a somewhat more systematicmanner. The original theory goes back to deBroglie and D. Bohm (see e.g. [94,95, 128, 129, 154, 471, 472, 532]) and in its modern version the dominantthemes seem to be contained in [88, 102, 288, 295, 324, 325, 326, 327, 328,329, 387, 402, 414, 415, 927, 948] with variations as in [110, 186, 187,188, 189, 191, 194, 195, 196, 197, 198, 346, 347, 373, 374, 375] basedon work of Bertoldi, Faraggi, and Matone (cf. also [68, 138, 148, 165, 236,305, 520, 574, 575, 576, 873, 881]) and cosmology following [123, 188, 189,219, 498, 499, 500, 501, 571, 709, 710, 711, 840, 841, 871, 872, 873,875, 876, 895, 989, 990]. In any event the quantum potential does enter intoany trajectory theory of deBroglie-Bohm (dBB) type. The history is discussedfor example in [471] (cf. also [68, 126, 127, 129, 154]) and we have seenhow this quantum potential idea can be formulated in various ways in terms ofstatistical mechanics, hydrodynamics, information and entropy, etc. when dealingwith different versions and origins of the SE. Given the existence of particles wefinds the pilot wave of thinking very attractive, with the wave function serving tochoreograph the particle motion (or perhaps to “create” particles and/or spacetimepaths). However the existence of particles itself is not such an assured matter andin field theory approaches for example one will deal with particle currents (cf. [701]and see also e.g. [94, 95, 326, 402, 948]). The whole idea of quantum particlepath seems in any case to be either fractal (cf. [1, 3, 14, 15, 186, 223, 232, 273,676, 715, 717, 720, 733, 734, 735, 736], stochastic (see e.g. [68, 148, 186,381, 382, 446, 447, 448, 449, 534, 536, 671, 674, 698, 805, 806], or fieldtheoretic (cf. [94, 95, 326, 402, 701, 702, 703, 704, 705, 706, 707, 948]. Thefractal approach sometimes imagines an underlying micro-spacetime where pathsare perhaps fractals with jumps, etc. and one possible advantage of a field theoreticapproach would be to let the fields sense the ripples, which as e.g. operator valuedSchwartz type distributions, they could well accomplish. In fact what comes intoquestion here is the structure of the vacuum and/or of spacetime itself. Onecan envision microstructures as in [186, 422, 676, 690] for example, textures(topological defects) as in [71, 74, 75, 170, 978], Planck scale structure andQFT, along with space-time uncertainty relations as in [71, 316, 317, 604, 1008],vacuum structures and conformal invariance as in [668, 669, 835, 831, 837,838], pilot wave cosmology as in [834, 881], ether theories as in [851, 919], etc.Generally there seems to be a sense in which particles cannot be measured as such

39

40 2. DEBROGLIE-BOHM IN VARIOUS CONTEXTS

and hence the idea of particle currents (perhaps corresponding to fuzzy particles orergodic clumps) should prevail perhaps along with the idea of probability packets.A number of arguments work with a (representative) trajectory as if it were asingle particle but there is no reason to take this too seriously; it could be thoughtof perhaps as a “typical” particle in a cloud but conclusions should perhaps alwaysbe constructed from an ensemble point of view. We will try to develop some of thisbelow. The sticky point as we see it now goes as follows. Even though one canwrite stochastic equations for (typical) particle motion as in the Nelson theoryfor example one runs into the problem of ever actually being able to localize aparticle. Indeed as indicated in [316, 317] (working in a relativistic context butthis should hold in general) one expects space time uncertainty relations even at asemiclassical level since any localization experiment will generate a gravitationalfield and deform spacetime. Thus there are relations [qµ, qν ] = iλ2

P Qµν where λP

is the Planck length and the picture of spacetime as a local Minkowski manifoldshould break down at distances of order λP . One wants the localization experimentto avoid creating a black hole (putting the object out of “reach”) for exampleand this suggests ∆x0(

∑31 ∆xi) λ2

P with ∆x1∆x2 + ∆x2∆x3 + ∆x3∆x1 λ2P

(cf. [316, 317]). On the other hand in [701] it is shown that in a relativisticbosonic field theory for example one can speak of currents and n-particle wavefunctions can have particles attributed to them with well defined trajectories,even though the probability of their experimental detection is zero. Thus oneenters an arena of perfectly respectible but undetectible particle trajectories. Thediscussion in [256, 326, 920, 953, 961] is also relevant here; some recourse tothe idea of beables, reality, and observables as beables, etc. is also involved (cf.[94, 95, 256, 961]). We will have something to say about all these matters.

The dominant approach as in [324, 325, 326, 327, 402, 948] will be discussedas needed (a thorough discussion would take a book in itself) and we only notehere that one is obliged to use the form ψ = Rexp(iS/) to make sense out of theconstructions (this is no problem with suitable provisos, e.g. that S is not constant- cf. [110, 191, 346, 347, 373, 374] and comments later). This leads to

() St +(S′)2

2m−(

2

2m

)(R′′

R

)+ V = 0; ∂tR

2 + ∂

(R2S′

m

)= 0

(cf. (1.1.1)) where Q = −2R′′/2mR arising from a SE i∂tψ = −(2/2m)ψxx +V ψ (we use 1-D for simplicity here). In [324] one emphasizes configurations basedon coordinates whose motion is choreographed by the SE according to the rule

() q = v =

mψ∗ψ′

|ψ|2 =

m(

ψ′

ψ

)The argument for () is based on obtaining the simplest Galilean and timereversal invariant form for velocity, transforming correctly under velocity boosts.This leads directly to () so that Bohmian mechanics (BM) is governed by() and the SE. It’s a fairly convincing argument and no recourse to Floydianneed be involved (cf. [110, 191, 347, 373, 374]). Note however that if S = cthen q = v = (/m)(R′/R) = 0 while p = S′ = 0 so this formulation seemsto avoid the S = constant problems indicated in [110, 191, 347, 373, 374].

1. THE KLEIN-GORDON AND DIRAC EQUATIONS 41

What makes the constant /m in () important here is that with this value theprobability density |ψ|2 on configuration space is equivariant. This means that viathe evolution of probability densities ρt + div(vρ) = 0 (as in (1.1.5)) the densityρ = |ψ|2 is stationary relative to ψ, i.e. ρ(t) retains the form |ψ(q, t)|2. One callsρ = |ψ|2 the quantum equilibrium density (QEDY) and says that a system is inquantum equilibrium when its coordinates are randomly distributed according tothe QEDY. The quantum equilibrium hypothesis (QEHP) is the assertion thatwhen a system has wave function ψ the distribution ρ of its coordinates satisfiesρ = |ψ|2.

1. THE KLEIN-GORDON AND DIRAC EQUATIONS

Before embarking on further discussion of QM it is necessary to describe someaspects of quantum field theory (QFT) and in particular to give some foundationfor the Klein-Gordon (KG) and Dirac equations. For QFT we rely on [120, 457,528, 764, 827, 1015] and concentrate on aspects of general quantum theory thatare expressed through such equations. We alternate between signature (−,+,+,+)and (+,−,−,−) in Minkowski space, depending on the source. It is hard to avoidusing units = c = 1 when sketching theoretical matters (which is personallyrepugnant) but we will set = c = 1 and shift to the general notation wheneverany real meaning is desired. Thus |length| ∼ |time| ∼ |energy|−1 ∼ |mass|−1

and m = the inverse Compton wavelength (mc/ = −1C ). The approaches in

[457, 764] seem best adapted to our needs and in particular [457] gives a nicediscussion motivating second quantization of a nonrelativistic SE (cf. [701] forfirst quantization). The resulting second quantization would be Galiean invariantbut not Lorentz invariant so we go directly to the KG equation as follows. Notethat there are often notational differences in various treatments of QFT and weuse that of [457] in general. Start now from E2 = p2+m2 (which is the relativisticform of E = p2/2m) to arrive, via E → i∂t and pj → −i∂j = −i∂/∂xj , at the KGequation

(1.1) (∂2t −∇2)φ + m2φ = 0

where φ = φ(x, t) is a scalar wave function. This can also be derived from anaction

(1.2) S(φ) =∫

d4xL(φ, ∂µφ) =12

∫d4x(∂µ∂µφ−m2φ)

(x0 = t, x = (x, t)), provided φ transforms as a Lorentz scalar (required also in(1.1)). The first problems arise from negative energy solutions (e.g. exp[i(k·x+ωt)]is a solution of (1.1) with E = −ω = −(k2 + m2)1/2). Secondly the energyspectrum is not bounded below (i.e. one could extract an arbitrary amount ofenergy from a single particle system). Further, using a positve square root ofE2 = p2 + m2 would involve a square root of a differential operator and nonlocalterms. Next observe that conserved currents jµ (with ∂µjµ = 0) arise a la E.Noether in the form

(1.3) j0 = ρ =i

2m(φ∗φt − φ∗

t φ); ji =1

2im(φ∗∂iφ− (∂iφ

∗)φ)

(where normal ordering is implicit here in order to avoid dealing with a vacuumenergy term - to be discussed later). We note that for the plane wave solution aboveρ = −ω/m = −(1/m)(k2 +m2)1/2 and this is not a good probability density. Thedifficulties are resolved by giving up the idea of a one particle theory; it is notcompatible with Lorentz invariance and the solution is to quantize the field φ.

Thus take S(φ) as in (1.2) with π = ∂L/∂(∂µφ) = ∂tφ = φ and construct aHamiltonian

(1.4) H =12

∫d3x[π2(x) + |∇φ(x)|2 + m2φ2(x)]

In analogy with QM where [x, p] = i one stipulates

(1.5) [φ(x, t), π(y, t)] = iδ(x− y); [φ(x, t), φ(y, t)] = [π(x, t), π(y, t)] = 0

The operator equation φ = i[H,φ] then yields π = φ and π = i[H,π] reproduces theKG equation. This is now a quantum field theory and for a particle interpretationone expands φ(x, t) in terms of classical solutions of the KG equation via

(1.6) φ(x, t) =∑

a(k)φ+k (x) + b(k)φ−

k (x) =

=∫

d3k(2π)3

12ωk

(a(k)e−i[ωkt−k·x] + b(k)ei[ωkt−k·x])

where ωk = (k2 + m2)1/2 and φ±k denotes a classical positive (resp. negative)

energy plane wave solution of (1.1) (k ·x = k0x0−k ·x = ωkt−k ·x). With φ(x, t)an operator one has operators a(k) and b(k); further since φ(x, t) is classically areal field we must have a Hermitian operator here and hence b(k) = a†(k). Thenormalization factor 1/2ωk is chosen for Lorentz invariance (cf. [457] for details).It follows immediately from π = ∂tφ that

(1.7) π(x, t) =∫

k3k

(2π)31

2ωk(−iωka(k)e−ik·x + iωka†(k)eik·x)

Some calculation (via Fourier formulas) leads then to

(1.8) a(k) =∫

d3xeik·x[ωkφ(x, t) + iπ(x, t)]

and the algebra of a, a† is then determined by [a(k), a†(k′)] = (2π)32ωkδ3(k−k′).The Hamiltonian (1.4) yields

(1.9) H =12

∫d3k

(2π)31

2ωkωk[a†(k)a(k)− a(k)a†(k)]

There is a bit of hocus-pocus here since the calculation gives a†a + (1/2)[a, a†] (=(1/2)(a†a + aa†) formally) but [a, a†] ∼ δ(0) corresponds to the sum oveer allmodes of zero point energies ωk/2. This infinite energy cannot be detected exper-imaentally since experiments only measure differences from the ground state of H.In any event the zero point field (ZPF) will be discussed in some detail later.

Now the ground state is defined via a(k)|0 >= 0 with < 0|0 >= 1 (a†(k)|0 >is a one particle state with energy ωk and momentum k while (a†(k))2|0 > con-tains two such particles, etc.). One notes however that the state a†(k)|0 > is not

normalizable since < 0|a(k)a†(k)|0 >= δ(0) is not normalizable. This is not sur-prising since a†(k) creates a particle of definite energy and momentum and by theuncertainty principle its location is unknown. Thus its wave function is a planewave and such states are not normalizable. In fact a†(k) is an operator valueddistribution and one can do calculations by “smearing” and considering states∫

d3kf(k)a†|0 > for functions f such that∫

d3k|f(k)|2 < ∞ for example. Onesees also that the bare vacuum |0 > is an eigenstate of the Hamiltonian but its en-ergy is divergent via < 0|H|0 >= (1/2)

∫d3kωkδ3(0) (where (2π)3δ3(0) ∼

∫d3x).

To deal with such infinities one subtracts them away, i.e. H → H− < 0|H|0 >and this corresponds to normal ordering the Hamiltonian via : aa† :=: a†a := a†aleading to

(1.10) : H :=∫

d3k

(2π)31

2ωkωka†(k)a(k)

with vanishing vacuum expectation.

REMARK 2.1.1. Regarding Lorentz invariance one recalls that the Lorentzgroup O(3, 1) is the set of 4× 4 matrices leaving the form s2 = (x0)2 −

∑(xi)2 =

xµgµνxν invariant. One writes (x′)µ = Λµνxν and notes that gµ = Λρ

µgρσΛσν ∼ g =

ΛT gΛ. Since s2 can be plus or minus there is a splitting into regions (x− y)2 > 0(time-like), (x − y)2 < 0 (space-like), and (x − y)2 = 0 (light-like). A standardparametrization for Lorentz boosts involves (x0 = ct)

(1.11) x′ =x + vt√

1− (v/c)2; y′ = y; z′ = z; t′ =

t + (vx/c2)√1− (v/c)2

One writes e.g. γ = 1/√

1− (v/c)2 = cosh(φ) with sinh(φ) = βγ = vγ/c.

REMARK 2.1.2. The total 4-momentum operator is

(1.12) Pµ =∫

d3k

(2π)31

2ωkkµa†(k)a(k)

and the total angular momentum operator is

(1.13) Mµν =∫

d3x(xµpν − xνpµ)

The Lorentz algebra (for infinitesimal Lorentz transformations) is

(1.14) [Mµν ,Mλσ] = i(ηµλMνσ − ηνλMµσ − ηµσMνλ + ηνσMµλ)

where ηµν = diag(1,−1,−1,−1).

REMARK 2.1.3. The commutator rules (1.5) are not manifestly Lorentzcovariant. However one can verify that the same quantum theory is obtainedregardless of what Lorentz frame is chosen; to do this one shows that the QMoperator forms of the Lorentz generators satisfy the Lorentz algebra after quanti-zation (this is given as an exercise in [457]).

EXAMPLE 1.1. Quantum fields are also discussed briefly in [471] and weextract here from this source. The approach follows [128] and one takes L =(1/2)∂µψ∂µψ = (1/2)[ψ2 − (∇ψ)2] as Lagrangian where ψ = ∂tψ and variational

technique yields the wave equation ψ = 0 ( = c = 1). Define conjugate mo-mentum as π = ∂L/∂ψ, the Hamiltonian via H = πψ − L = (1/2)[π2 + (∇ψ))2],and the field Hamiltonian by H =

∫Hd3x. Replacing π by δS/δψ where S[ψ] is a

functional the classical HJ equation of the field ∂tS + H = 0 becomes

(1.15)∂S

∂t+

12

∫d3x

[(δS

δψ

)2

+ (∇S)2]

= 0

The term (1/2)∫

d3x(∇ψ)2 plays the role of an external potential. To quantizethe system one treats ψ(x) and π(x) as Schrodinger operators with [ψ(x), ψ(x′)] =[π(x), π(x′)] = 0 and [ψ(x), π(x′)] = iδ(x−x′). Then one works in a representation|ψ(x) > in which the Hermitian operator ψ(x) is diagonal. The Hamiltonianbecomes an operator H acting on a wavefunction Ψ[ψ(x), t) =< ψ(x)|Ψ(t) >which is a functional of the real field ψ and a function of t. This is not a pointfunction of x since Ψ depends on the variable ψ for all x. Now the SE for the fieldis i∂tΨ = HΨ or explicitly

(1.16) i∂Ψ∂t

=∫

d3x12

[− δ2

δψ2+ (∇ψ)2

]Ψ

Thus ψ is playing the role of the space variable x in the particle SE and thecontinuous index x here is analogous to a discrete index n in the many particletheory. To arrive at a causal interpretation now one writes Ψ = Rexp(iS) forR,S[ψ, t] real functionals and decomposes (1.16) as

(1.17)∂S

∂t+

12

∫d3x

[(δS

δψ

)2

+ (∇ψ)2]+Q = 0;

∂R2

∂t+∫

d3xδ

δψ

(R2 δS

δψ

)= 0

where the quantum potential is now Q[ψ, t] = −(1/2R)∫

d3x(δ2R/δψ2). (1.17)now gives a conservation law wherein, at time t, R2Dψ is the probability for thefield to lie in an element of volume Dψ around ψ, where Dψ means roughly

∏x dψ

and there is a normalization∫|Ψ|2Dψ = 1. Now introduce the assumption that at

each instant t the field ψ has a well defined value for all x as in classical field theory,whatever the state Ψ. Then the time evolution is obtained from the solution ofthe “guidance” formula

(1.18)∂ψ(x, t)

∂t=

δS[ψ(x), t]δψ(x)

∣∣∣∣ψ(x)=ψ(x,t)

(analogous to mx = ∇S) once one has specified the initial function ψ0(x) in theHJ formalism. To find the equation of motion for the field coordinates apply δ/δψ

to the HJ equation (1.17) to get formally (ψ ∼ δS/δψ)

(1.19)d

dtψ = − δ

δψ

[Q +

12

∫d3x(∇ψ)2

];

d

dt=

∂

∂t+∫

d3x∂ψ

∂t

δ

δψ

This is analogous to mx = −∇(V +Q) and, noting that dψ/dt = ∂ψ/dt and takingthe classical external force term to the right one arrives, via standard variational


methods, at

(1.20) ψ(x), t) = − δQ[ψ(x, t]δψ(x)

∣∣∣∣ψ(x)=ψ(x,t)

(note (δ/δψ)∫

d3x(∇ψ)2 ∼ −2∆ψ and (δ/δψ0∂tψ = ∂t(δ/δψ)ψ = 0). The quan-tum force term on the right side is responsible for all the characteristic effects ofQFT. In particular comparing to a classical massive KG equation ψ + m2ψ = 0with suitable initial conditions one can argue that the quantum force generatesmass in the sense that the massless quantum field acts as if it were a classical fieldwith mass given via the quantum potential (cf. Remark 2.2.1 below).

1.1. ELECTROMAGNETISM AND THE DIRAC EQUATION. Itwill be useful to have a differential form discription of EM fields and we supply thisvia [723]. Thus one thinks of tensors T = T σ

µν∂σ ⊗ dxµ ⊗ dxν with contractionsof the form T (dxσ, ∂σ) ∼ Tνdxν . For η = ηµνdxµ⊗ dxν one has η−1 = ηµν∂µ⊗ ∂ν

and ηη−1 = 1 ∼ diag(δµµ). Note also e.g.

(1.21) ηµνdxµ ⊗ dxν(u,w) = ηµνdxµ(u)dxν(w) =

= ηµνdxµ(uα∂α)dxν(wτ∂τ ) = ηµνuµwν

(1.22) η(u) = ηµνdxµ ⊗ dxν(u) = ηµνdxµ(u)dxν =

= ηµνdxµ(uα∂α)dxν = ηµνuµdxν = uνdxν

for a metric η. Recall α ∧ β = α⊗ β − β ⊗ α and

(1.23) α ∧ β = αµdxµ ∧ βνdxν = (1/2)(αµβν − ανβµ)dxµ ∧ dxν

The EM field tensor is F = (1/2)Fµνdxµ ∧ dxν where

(1.24) Fµν =

⎛⎜⎜⎝0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

⎞⎟⎟⎠ ;

F = Exdx0 ∧ dx1 + Eydx0 ∧ dx2 + Ezdx0 ∧ dx3 −Bzdx1 ∧ dx2+

+Bydx1 ∧ dx3 −Bxdx2 ∧ dx3

The equations of motion of an electric charge is then dp/dτ = (e/m)F(p) wherep = pµ∂µ. There is only one 4-form, namely ε = dx0 ∧ dx1 ∧ dx2 ∧ dx3 =(1/4!)εµνστdxµ∧dxν ∧dxσ ∧dxτ where εµνστ is totally antisymmetric. Recall alsofor α = αµν···dxµ ∧ dxν · · · one has dα = dαµν··· ∧ dxµ ∧ dxν · · · = ∂ααµν···dxσ ∧dxµ ∧ dxν · · · and ddα = 0. Define also the Hodge star operator on F and j via∗F = (1/4)εµνστF στdxµ ∧ dxν and ∗j = (1/3!)εµνστ jτdxµ ∧ dxν ∧ dxσ; these arecalled dual tensors. Now the Maxwell equations are

(1.25) ∂µFµν =4π

cjν ; ∂αFµν + ∂µF να + ∂νFαµ = 0

and this can now be written in the form

(1.26) dF = 0; d∗F =4π

c∗ j


and 0 = d∗j = 0 is automatic. In terms of A = Aµdxµ where F = dA the relationdF = 0 is an identity ddA = 0.

A few remarks about the tensor nature of jµ and Fµν are in order and wewrite n = n(x) and v = v(x) for number density and velocity with charge densityρ(x) = qn(x) and current density j = qn(x)v(x). The conservation of particlenumber leads to ∇ · j + ρt = 0 and one writes

(1.27) jν = (cρ, jx, jy, jz) = (cρn, qnvx, qnvy, qnvz) ≡ jν = n0quν ≡ jν = ρ0u

ν

where n0 = n√

1− (v2/c2) and ρ0 = qn0 (ρ0 here is charge density). Since jν con-sists of uν multiplied by a scalar it must have the transformation law of a 4-vectorj′β = aβ

νjν under Lorentz transformations (aβν ∼ Λβ

ν ). Then the conservation lawcan be written as ∂νjν = 0 with obvious Lorentz invariance. After some argumentone shows also that Fµν = a ν

β a µα F

′αβ under Lorentz transformations so Fµν isindeed a tensor. The equation of motion for a charged particle can be written nowas

(1.28) (dp/dt) = qE + (q/c)v ×B; p = mv/√

1− (v2/c2)

This is equivalent to dpµ/dt = (q/m)pνFµν with obvious Lorentz invariance. Theenergy momentum tensor of the EM field is

(1.29) Tµν = −(1/4π)[FµαF να − (1/4)ηµνFαβFαβ ]

(cf. [723] for details) and in particular T 00 = (1/8π)(E2 +B2) while the Poyntingvector is T 0k = (1/4π)(E×B)k.

One can equally well work in a curved space where e.g. covariant derivativesare defined via ∇nT = limdλ→0[(T (λ+dλ)−T (λ)−δT ]/dλ where δT is the changein T produced by parallel transport. One has then the usual rules ∇u(T ⊗ R) =∇uT ⊗R+T ⊗∇uR and for v = vν∂ν one finds ∇µv = ∂µvν∂ν + vν∇µ∂ν . Now ifv was constructed by parallel transport its covariant derivative is zero so, actingwith the dual vector dxα gives

(1.30)∂xν

∂xµdxα(∂ν) + vνdxα(∇µ∂ν) = 0 ≡ ∂µvα + vνdxα(∇µ∂ν) = 0

Comparing this with the standard ∂µvα + Γαµνvν = 0 gives dxα(∇µ∂ν) = Γα

µν .One can show also for vectors u, v, w (boldface omitted) and a 1-form α

(1.31) (∇u∇v −∇v∇u − uv + vu)α(w) = R(α, u, v, w);

R = F σβµν∂σ ⊗ dxβ ⊗ dxµ ⊗ dxν

so R represents the Riemann tensor.

For the nonrelativistic theory we recall from [649] that one can define a trans-verse and longitudinal component of a field F via

(1.32) F ||(r) = − 14π

∫d3r′

∇′ · F (r′)|r − r′| ; F⊥(r) =

14π∇×∇×

∫d3r′

F (r′)|r − r′|

For a point particle of mass m and charge e in a field with potentials A and φone has nonrelativistic equations mx = eE + (e/c)v × B (boldface is suppressed


here) where one recalls B = ∇ × A, v = x, and E = −∇φ − (1/c)At with H =(1/2m)(p− (e/c)A)2 + eφ leading to

(1.33) x =1

2m

(p− e

cA)

; p =e

c[v ×B + (v · ∇)A]− e∇φ

Recall here also

(1.34) B = ∇×A; ∇ · E = 0; ∇ ·B = 0; ∇× E = −(1/c)Bt;

∇×B = (1/c)Et; E = −(1/c)At −∇φ

(the Coulomb gauge∇·A = 0 is used here). One has now E = E⊥+E|| ∼ ET +EL

with ∇ · E⊥ = 0 and ∇ × E|| = 0 and in Coulomb gauge E⊥ = −(1/c)At andE|| = −∇φ. Further

(1.35) H ∼ 12m

(p− e

cA)2

+ eφ +18π

∫d3r((E⊥)2 + B2)

(covering time evolution of both particle and fields).

For the relativistic theory one goes to the Dirac equation

(1.36) i(∂t + α · ∇)ψ = βmψ

which, to satisfy E2 = p2 +m2 with E ∼ i∂t and p ∼ −i∇, implies −∂2t ψ = (−iα ·

∇+βm)2ψ and ψ will satisfy the KG equation if β2 = 1, αiβ+βαi ≡ αi, β = 0,and αi, αj = 2δij (note c = = 1 here with α · ∇ ∼

∑αµ∂µ and cf. [647, 650]

for notations and background). This leads to matrices

(1.37) σ1 =(

0 11 0

); σ2 =

(0 −ii 0

);

σ3 =(

1 00 −1

); αi =

(0 σi

σi 0

); β =

(1 00 −1

)where αi and β are 4 × 4 matrices. Then for convenience take γ0 = β and γi =βαi which satisfy γµ, γν = 2gµν (Lorentz metric) with (γi)† = −γi, (γi)2 =−1, (γ0)† = γ0, and (γ0)2 = 1. The Dirac equation for a free particle can now bewritten

(1.38)(

iγµ ∂

∂xµ−m

)ψ = 0 ≡ (i∂/−m)ψ = 0

where A/ = gµνγµAν = γµAµ and ∂/ = γµ∂µ. Taking Hermitian conjugates in(1.36), noting that α and β are Hermitian, one gets ψ(i

←−∂/ +m) = 0 where ψ = ψ†β.

To define a conserved current one has an equation ψγµ∂µψ+γµψµψ = ∂µ(ψγµψ) =0 leading to the conserved current jµ = ψγµψ = (ψ†ψ, ψ†αψ) (this means ρ = ψ†ψand j = ψ†αψ with ∂tρ+∇· j = 0). The Dirac equation has the Hamiltonian form

(1.39) i∂tψ = −iα · ∇ψ + βmψ = (α · p + βm)ψ ≡ Hψ

(α · p ∼∑

αµpµ). To obtain a Dirac equation for an electron coupled to a pre-scribed external EM field with vector and scalar potentials A and φ one substitutespµ → pµ − eAµ, i.e. p→ p− eA and p0 = i∂t → i∂t − eΦ, to obtain

(1.40) i∂tψ = [α · (p− eA) + eΦ + βm]ψ


This identifies the Hamiltonian as H = α · (p−eA)+eΦ+βm = α ·p+βm+Hint

where Hint = −eα · A + eΦ, suggesting α as the operator corresponding to thevelocity v/c; this is strengthened by the Heisenberg equations of motion

(1.41) r =(

1i

)[r,H] = α; π =

(1i

)[π,H] = e(E + α×B)

Another bit of notation now from [650] is useful. Thus (again with c = = 1)one can define e.g.

(1.42) σz = −iαxαy; σx = −iαyαz; σy = −iαzαx; ρ3 = β;

ρ1 = σzαz = −iαxαyαz; ρ2 = iρ1ρ3 = βαxαyαz

so that β = ρ3 and αk = ρ1σk. Recall also that the angular momentum of a parti-

cle is = r×p (∼ (−i)r×∇) with components k satisfying [x, y] = iz, [y, z] =ix, and [z, x] = iy. Any vector operator L satisfying such relations is called anangular momentum. Next one defines σµν = (1/2)i[γµ, γν ] = iγµγν (µ = ν) andSαβ = (1/2)σαβ . Then the 6 components Sαβ satisfy

(1.43) S10 = (i/2)αx; S20 = (i/2)αy; S30 = (i/1)αz;

S23 = (1/2)σx; , S31 = (1/2)σy; S12 = (1/2)σz

The Sαβ arise in representing infinitesimal rotations for the orthochronous Lorentzgroup via matrices I + iεSαβ . Further one can represent total angular momentumJ in the form J = L + S where L = r × p and S = (1/2)σ (L is orbital angularmomentum and S represents spin). We recall that the gamma matrices are givenvia γ = βα. Finally [(i∂t − eφ)− α · (−i∇− eA)− βm]ψ = 0 (cf. (1.40)) and onegets

(1.44) [iγµDµ −m)ψ = [γµ(i∂µ − eAµ)−m]ψ = 0

Dµ = ∂µ + ieAµ ≡ (∂0 + ieφ,∇− ieA)Working on the left with (−iγλDλ − m) gives then [γλγµDλDµ + m2]ψ = 0where γλγµ = gλµ + (1/2)[γλ, γµ]. By renaming the dummy indices one obtains[γλ, γµ]DλDµ = −[γλ, γµ]DµDλ = (1/2)[γλ, γµ][Dλ, Dµ] leading to

(1.45) [Dλ, Dµ] = ie[∂λ, Aµ] + ie[Aλ, ∂µ] = ie(∂λAµ − ∂µAλ) = ieFλµ

This yields then γλγµDλDµ = DµDλ + eSλµFλµ where Sλµ represents the spin ofthe particle. Therefore one can write [DµDµ + eSλµFλµ + m2]ψ = 0. Comparingwith the standard form of the KG equation we see that this differs by the termeSλµFλµ which is the spin coupling of the particle to the EM field and has noclassical analogue.

2. BERTOLDI-FARAGGI-MATONE THEORY

The equivalence principle (EP) of Faraggi-Matone (cf. [110, 191, 193, 198,347, 641]) is based on the idea that all physical systems can be connected bya coordinate transformation to the free situation with vanishing energy (i.e. allpotentials are equivalent under coordinate transformations). This automaticallyleads to the quantum stationary Hamilton-Jacobi equation (QSHJE) which is athird order nonlinear differential equation providing a trajectory representation of

2. BERTOLDI-FARAGGI-MATONE THEORY 49

quantum mechanics (QM). The theory transcends in several respects the Bohmtheory and in particular utilizes a Floydian time (cf. [373, 374]) leading toq = p/mQ = p/m where mQ = m(1 − ∂EQ) is the “quantum mass” and Qthe “quantum potential” (cf. also Section 7.4). Thus the EP is reminscient ofthe Einstein equivalence of relativity theory. This latter served as a midwife tothe birth of relativity but was somewhat inaccurate in its original form. It isbetter put as saying that all laws of physics should be invariant under generalcoordinate transformations (cf. [723]). This demands that not only the formbut also the content of the equations be unchanged. More precisely the equa-tions should be covariant and all absolute constants in the equations are to be leftunchanged (e.g. c, , e, m and ηµν = Minkowski tensor). Now for the EP, theclassical picture with Scl(q, Q0, t) the Hamilton principal function (p = ∂Scl/∂q)and P 0, Q0 playing the role of initial conditions involves the classical HJ equation(CHJE) H(q, p = (∂Scl/∂q), t)+(∂Scl/∂t) = 0. For time independent V one writesScl = Scl

0 (q, Q0)−Et and arrives at the classical stationary HJ equation (CSHJE)(1/2m)(∂Scl

0 /∂q)2 +W = 0 where W = V (q)−E. In the Bohm theory one lookedat Schrodinger equations iψt = −(2/2m)ψ′′ + V ψ with ψ = ψ(q)exp(−iEt/)and ψ(q) = R(qexp(iW /) leading to

(2.1)(

12m

)(W ′)2 + V − E − 2R′′

2mR= 0; (R2W ′)′ = 0

where Q = −2R′′/2mR was called the quantum potential; this can be writtenin the Schwartzian form Q = (2/4m)W ; q (via R2W ′ = c). Here f ; q =(f ′′′/f ′)−(3/2)(f ′′/f ′)2. Writing W = V (q)−E as in above we have the quantumstationary HJ equation (QSHJE)

(2.2) (1/2m)(∂W ′/∂q)2 + W(q) + Q(q) = 0 ≡W = −(2/4m)exp(2iS0/); q

This was worked out in the Bohm school (without the Schwarzian connections) butψ = Rexp(iW /) is not appropriate for all situations and care must be taken (W =constant must be excluded for example - cf. [347, 373, 374]). The technique ofFaraggi-Matone (FM) is completely general and with only the EP as guide oneexploits the relations between Schwarzians, Legendre duality, and the geometry ofa second order differential operator D2

x + V (x) (Mobius transformations play animportant role here) to arrive at the QSHJE in the form

(2.3)1

2m

(∂Sv

0 (qv)∂qv

)2

+ W(qv) + Qv(qv) = 0

where v : q → qv represents an arbitrary locally invertible coordinate transforma-tion. Note in this direction for example that the Schwarzian derivative of the theratio of two linearly independent elements in ker(D2

x + V (x)) is twice V (x). Inparticular given an arbitrary system with coordinate q and reduced action S0(q)the system with coordinate q0 corresponding to V −E = 0 involves W(q) = (q0; q)where (q0, q) is a cocycle term which has the form (qa; qb) = −(2/4m)qa; qb. Infact it can be said that the essence of the EP is the cocycle condition

(2.4) (qa; qc) = (∂qcqb)2[(qa; qb)− (qc; qb)]


In addition FM developed a theory of (x, ψ) duality (cf. [346])) which relatedthe space coordinate and the wave function via a prepotential (free energy) inthe form F = (1/2)ψψ + iX/ε for example. A number of interesting philosophicalpoints arise (e.g. the emergence of space from the wave function) and we connectedthis to various features of dispersionless KdV in [191, 198] in a sort of extendedWKB spirit (cf. also Section 7.3). One should note here that although a formψ = Rexp(iW /) is not generally appropriate it is correct when one is dealingwith two independent solutions of the Schrodinger equation ψ and ψ which are notproportional. In this context we utilized some interplay between various geometricproperties of KdV which involve the Lax operator L2 = D2

x + V (x) and of coursethis is all related to Schwartzians, Virasoro algebras, and vector fields on S1 (seee.g. [191, 192, 198, 200, 201]). Thus the simple presence of the Schrodingerequation (SE) in QM automatically incorporates a host of geometrical properties ofDx = d/dx and the circle S1. In fact since the FM theory exhibits the fundamentalnature of the SE via its geometrical properties connected to the QSHJE one couldspeculate about trivializing QM (for 1-D) to a study of S1 and ∂x.

We import here some comments based on [110] concerning the Klein-Gordon(KG) equation and the equivalence principle (EP) (details are in [110] and cf.also [164, 165, 166, 298, 472, 474, 473, 478, 479, 480, 666, 667] for theKG equation which is treated in some detail later at several places in this book).One starts with the relativistic classical Hamilton-Jacobi equation (RCHJE) witha potential V (q, t) given as

(2.5)1

2m

D∑1

(∂kScl(q, t))2 + Wrel(q, t) = 0;

Wrel(q, t) =1

2mc2[m2c4 − (V (q, t) + ∂tS

cl(q, t))2]

In the time-independent case one has Scl(q, t) = Scl0 (q)− Et and (2.3) becomes

(2.6)1

2m

D∑1

(∂kScl0 )2 + Wrel = 0; Wrel(q) =

12mc2

[m2c4 − (V (q)− E)2]

In the latter case one can go through the same steps as in the nonrelativistic caseand the relativistic quantum HJ equation (RQHJE) becomes

(2.7) (1/2m)(∇S0)2 + Wrel − (2/2m)(∆R/R) = 0; ∇ · (R2∇S0) = 0

these equations imply the stationary KG equation

(2.8) −2c2∆ψ + (m2c4 − V 2 + 2EV − E2)ψ = 0

where ψ = Rexp(iS0/). Now in the time dependent case the (D+1)-dimensionalRCHJE is (ηµν = diag(−1, 1, · · · , 1)

(2.9) (1/2m)ηµν∂µScl∂νScl + W′rel = 0;

W′rel = (1/2mc2)[m2c4 − V 2(q)− 2cV (q)∂0S

cl(q)]with q = (ct, q1, · · · , qD). Thus (2.9) has the same structure as (2.6) with Euclideanmetric replaced by the Minkowskian one. We know how to implement the EP by


adding Q via (1/2m)(∂S)2 + Wrel + Q = 0 (cf. [347] and remarks above). Notenow that W′

rel depends on Scl requires an identification

(2.10) Wrel = (1/2mc2)[m2c4 − V 2(q)− 2cV (q)∂0S(q)]

(S replacing Scl) and implementation of the EP requires that for an arbitrary Wa

state (q ∼ qa) one must have

(2.11) Wbrel(q

b) = (pb|pa)Warel(q

a) + (qq; qb); Qb(qb) = (pb|pa)Q(qa)− (qa; qb)

where

(2.12) (pb|p) = [ηµνpbµpb

ν/ηµνpµpν ] = pT JηJT p/pT ηp; Jµν = ∂qµ/∂qbν

(J is a Jacobian and these formulas are the natural multidimensional generaliza-tion - see [110] for details). Furthermore there is a cocycle condition (qa; qc) =(pc|pb)[(qa; qb)− (qc; qb)].

Next one shows that Wrel = (2/2m)[(Rexp(iS/))/Rexp(iS/)] and hencethe corresponding quantum potential is Qrel = −(2/2m)[R/R]. Then theRQHJE becomes (1/2m)(∂S)2 + Wrel + Q = 0 with ∂ · (R2∂S) = 0 (here R =∂µ∂µR) and this reduces to the standard SE in the classical limit c → ∞ (note∂ ∼ (∂0, ∂1, · · · , ∂D) with q0 = ct, etc. - cf. (2.9)). To see how the EP is simplyimplemented one considers the so called minimal coupling prescription for an in-teraction with an electromagnetic four vector Aµ. Thus set P cl

µ = pclµ + eAµ where

pclµ is a particle momentum and P cl

µ = ∂µScl is the generalized momentum. Thenthe RCHJE reads as (1/2m)(∂Scl − eA)2 + (1/2)mc2 = 0 where A0 = −V/ec.Then W = (1/2)mc2 and the critical case W = 0 corresponds to the limit situa-tion where m = 0. One adds the standard Q correction for implementation of theEP to get (1/2m)(∂S − eA)2 + (1/2)mc2 + Q = 0 and there are transformationproperties (here (∂S − eA)2 ∼

∑(∂µS − eAµ)2)

(2.13) W(qb) = (pb|pa)Wa(qa) + (qa; qb); Qb(qb) = (pq|pa)Qa(qa)− (qa; qb)

(pb|p) =(pb − eAb)2

(p− eA)2=

(p− eA)T JηJT (p− eA)(p− eA)T η(p− eA)

Here J is a Jacobian Jµν = ∂qµ/∂qbν

and this all implies the cocycle conditionagain. One finds now that (recall ∂ · (R2(∂S − eA)) = 0 - continuity equation)

(2.14) (∂S − eA)2 = 2

(R

R− D2(ReiS/)

ReiS/

); Dµ = ∂µ −

i

eAµ

and it follows that

(2.15) W =2

2m

D2(ReiS/)ReiS/ ; Q = − 2

2m

R

R; D2 = − 2ieA∂

− e2A2

2− ie∂A

(2.16) (∂S − eA)2 + m2c2 − 2 R

R= 0; ∂ · (R2(∂S − eA)) = 0

Note also that (2.9) agrees with (1/2m)(∂Scl − eA)2 + (1/2)mc2 = 0 after settingWrel = mc2/2 and replacing ∂µScl by ∂µScl − eAµ. One can check that (2.16)implies the KG equation (i∂ + eA)2ψ + m2c2ψ = 0 with ψ = Rexp(iS/).


REMARK 2.2.1. We extract now a remark about mass generation and theEP from [110]. Thus a special property of the EP is that it cannot be implementedin classical mechanics (CM) because of the fixed point corresponding to W =0. One is forced to introduce a uniquely determined piece to the classical HJequation (namely a quantum potential Q). In the case of the RCHJE the fixed pointW(q0) = 0 corresponds to m = 0 and the EP then implies that all the other massescan be generated by a coordinate transformation. Consequently one concludes thatmasses correspond to the inhomogeneous term in the transformation properties ofthe W0 state, i.e. (1/2)mc2 = (q0; q). Furthermore by (2.13) masses are expressedin terms of the quantum potential (1/2)mc2 = (p|p0)Q0(q0)−Q(q). In particularin [347] the role of the quantum potential was seen as a sort of intrinsic self energywhich is reminiscent of the relativistic self energy and this provides a more explicitevidence of such an interpretation.

REMARK 2.2.2. In a previous paper [194] (working with stationary statesand ψ satisfying the Schrodinger equation (SE) −(2/2m)ψ′′ + V ψ = Eψ) wesuggested that the notion of uncertainty in quantum mechanics (QM) could bephrased as incomplete information. The background theory here is taken to bethe trajectory theory of Bertoldi-Faraggi-Matone (and Floyd) as above and theidea in [194] goes as follows. First recall that microstates satisfy a third orderquantum stationary Hamilton-Jacobi equation (QSHJE)

(2.17)1

2m(S′

0)2 + W(q) + Q(q) = 0; Q(q) =

2

4mS0; q;

W(q) = − 2

4mexp(2iS0/); q ∼ V (q)− E

where f ; q = (f ′′′/f ′) − (3/2)(f ′′/f ′)2 is the Schwarzian and S0 is the Hamil-ton principle function. Also one recalls that the EP of Faraggi-Matone can onlybe implemented when S0 = const; thus consider ψ = Rexp(iS0/) with Q =−2R′′/2mR and (R2S′

0)′ = 0 where S′

0 = p and mQq = p with mQ = m(1 −∂EQ) and t ∼ ∂ES0 (Q in (2.17) is the definitive form - cf. [349]). Thus mi-crostates require three initial or boundary conditions in general to determineS0 whereas the SE involves only two such conditions (cf. also Section 7.4 and[138, 140, 139, 305, 306, 307, 308, 309, 347, 348, 349, 373, 374, 375, 520]).Hence in dealing with the SE in the standard QM Hilbert space formulation oneis not using complete information about the “particles” described by microstatetrajectories. The price of underdetermination is then uncertainty in q, p, t forexample. In the present note we will make this more precise and add furtherdiscussion. Following [197] we now make this more precise and add further dis-cussion. For the stationary SE −(2/2m)ψ′′ + V ψ = Eψ it is shown in [347] thatone has a general formula

(2.18) e2iS0(δ)/ = eiα w + i

w − i

(δ ∼ (α, )) with three integration constants, α, 1, 2 where = 1 + i2 andw ∼ ψD/ψ ∈ R. Note ψ and ψD are linearly independent solutions of the SE and


one can arrange that ψD/ψ ∈ R in describing any situation. Here p is determinedby the two constants in and has a form

(2.19) p =±Ω1

|ψD − iψ|2

(where w ∼ ψD/ψ above and Ω = ψ′ψD − ψ(ψD)′). Now let p be determinedexactly with p = p(q, E) via the Schrodinger equation and S′

0. Then q = (∂Ep)−1

is also exact so ∆q = (∂Ep)−1(τ)∆t for some τ with 0 ≤ τ ≤ t is exact (up toknowledge of τ). Thus given the wave function ψ satisfying the stationary SEwith two boundary conditions at q = 0 say to fix uniqueness, one can create aprobability density |ψ|2(q, E) and the function S′

0. This determines p uniquelyand hence q. The additional constant needed for S0 appears in (2.18) and we canwrite S0 = S0(α, q, E) since from (2.18) one has

(2.20) S0 − (/2)α = −(i/2)log(β)

and β = (w + i)/(w − i) with w = ψD/ψ is to be considered as known via adetermination of suitable ψ, ψD. Hence ∂αS0 = −/2 and consequently ∆S0 ∼∂αS0δα = −(/2)∆α measures the indeterminacy or uncertainty in S0.

Let us expand upon this as follows. Note first that the determination ofconstants necessary to fix S0 from the QSHJE is not usually the same as thatinvolved in fixing , in (2.18). In paricular differentiating in q one gets

(2.21) S′0 = − iβ′

β; β′ = − 2iw′

(w − i)2

Since w′ = −Ω/ψ2 where Ω = ψ′ψD − ψ(ψD)′ we get β′ = −2i1Ω/(ψD − iψ)2

and consequently

(2.22) S′0 = − 1Ω

|ψD − iψ|2

which agrees with p in (2.19) (± simply indicates direction). We see that e.g.S0(x0) = i1Ω/|ψD(x0) − iψ(x0)|2 = f(1, 2, x0) and S′′

0 = g(1, 2, x0) de-termine the relation between (p(x0), p′(x0)) and (1, 2) but they are generallydifferent numbers. In any case, taking α to be the arbitrary unknown constantin the determination of S0, we have S0 = S0(q, E, α) with q = q(S0, E, α) andt = t(S0, E, α) = ∂ES0 (emergence of time from the wave function). One can thenwrite e.g.

(2.23) ∆q = (∂q/∂S0)(S0, E, α)∆S0 = (1/p)(q, E)∆S0 = −(1/p)(q, E)(/2)∆α

(for intermediate values (S0, q)) leading to

THEOREM 2.1. With p determined uniquely by two “initial” conditions sothat ∆p is determined and q given via (2.18) we have from (2.23) the inequality∆p∆q = O() which resembles the Heisenberg uncertainty relation.

COROLLARY 2.1. Similarly ∆t = (∂t/∂S0)(S0, E, α)∆S0 for some inter-mediate value S0 and hence as before ∆E∆t = O() (∆E being precise).


Note that there is no physical argument here; one is simply looking at thenumber of conditions necessary to fix solutions of a differential equation. In fact(based on some corresondence with E. Floyd) it seems somewhat difficult to pro-duce a viable physical argument. We refer also to Remark 3.1.2 for additionaldiscussion.

REMARK 2.2.3. In order to get at the time dependent SE from the BFM(Bertoldi-Faraggi-Matone) theory we proceed as follows. From the previous dis-cussion on the KG equation one sees that (dropping the EM terms) in the timeindependent case one has Scl(q, t) = Scl

0 (q)− Et

(2.24)

(1/2m)∑D

1 (∂kScl0 )2 + Wrel = 0; Wrel(q) = (1/2mc2)[m2c4 − (V (q)− E)2]

leading to a stationary RQHJE

(2.25) (1/2m)(∇S0)2 + Wrel − (2/2m)(∆R/R) = 0; ∇ · (R2∇S0) = 0

This implies also the stationary KG equation

(2.26) −2c2∆ψ + (m2c4 − V 2 + 2V E − E2)ψ = 0

Now in the time dependent case one can write (1/2m)ηµν∂µScl∂νScl + W′rel = 0

where η ∼ diag(−1, 1, · · · , 1) and

(2.27) W′rel(q) = (1/2mc2)[m2c4 − V 2(q)− 2cV (q)∂0S

cl(q)]

with q ≡ (ct, q1, · · · , qD). Thus we have the same structure as (2.24) with Euclid-ean metric replaced by a Minkowskian one. To implement the EP we have tomodify the classical equation by adding a function to be determined, namely(1/2m)(∂S)2+Wrel+Q = 0 ((∂S)2 ∼

∑(∂µS)2 etc.). Observe that since W′

rel de-pends on Scl we have to make the identification Wrel = (1/2mc2)[m2c4−V 2(q)−2cV (q)∂0S(q)] which differs from W′

rel since S now appears instead of Scl. Imple-mentation of the EP requires that for an arbitrary Wa state

(2.28) Wbrel(q

b) = (pb|pa)Warel(q

a) + (qq; qb); Qb(qb) = (pb|pa)Qa(qa)− (qa; qb)

where now (pb|p) = ηµνpbµpb

ν/ηµνpµpν = pT JηJT p/pT ηp and Jµν = ∂qµ/∂(qb)ν .

This leads to the cocycle condition (qa; qc) = (pc|pb)[(qq; qb) − (qc; qb)] as before.Now consider the identity

(2.29) α2(∂S)2 = (Rexp(αS))/Rexp(αS)− (R/R)− (α∂ · (R2∂S)/R2)

and if R satisfies the continuity equation ∂ · (R2∂S) = 0 one sets α = i/ to obtain

(2.30)1

2m(∂S)2 = − 2

2m

(ReiS/)ReiS/ +

2

2m

R

R

Then it is shown that Wrel = (2/2m)((Rexp(iS/))/Rexp(iS/) so Qrel =−(2/2m)(R/R). Thus the RQHJE has the form (cf. (2.14) - (2.16))

(2.31)1

2m(∂S)2 + Wrel −

2

2m

R

R= 0; ∂ · (R2∂S) = 0

3. FIELD THEORY MODELS 55

Now for the time dependent SE one takes the nonrelativistic limit of theRQHJE. For the classical limit one makes the usual substitution S = S′ −mc2tso as c →∞ Wrel → (1/2)mc2 + V and −(1/2m)(∂0S)2 → ∂tS

′ − (1/2)mc2 with∂(R2∂S) = 0→ m∂t(R′)2 +∇ · ((R′)2∇S′) = 0. Therefore (removing the primes)(2.31) becomes (1/2m)(∇S)2 + V + ∂tS − (2/2m)(∆R/R) = 0 with the timedependent nonrelativistic continuity equation being m∂tR

2 + ∇ · (R2∇S) = 0.This leads then (for ψ ∼ Rexp(iS/)) to the SE

(2.32) i∂tψ =(− 2

2m∆ + V

)ψ

One sees from all this that the BFM theory is profoundly governed by the equiva-lence principle and produces a usable framework for computation. It is surprisingthat it has not attracted more adherents.

3. FIELD THEORY MODELS

In trying to imagine particle trajectories of a fractal nature or in a fractalmedium we are tempted to abandon (or rather relax) the particle idea and switchto quantum fields (QF). Let the fields sense the bumps and fractality; if one canthink of fields as operator valued distributions for example then fractal supports forexample are quite reasonable. There are other reasons of course since the notion ofparticle in quantum field theory (QFT) has a rather fuzzy nature anyway. Thenof course there are problems with QFT itself (cf. [973]) as well as argumentsthat there is no first quantization (except perhaps in the Bohm theory - cf. [701,1016]). We review here some aspects of particles arising from QF and QFTmethods, especially in a Bohmian spirit (cf. [77, 110, 256, 324, 325, 326, 454,472, 478, 479, 480, 494, 532, 634, 701, 702, 703, 704, 705, 706, 707,708, 709, 710, 711, 984]). We refer to [454, 973] for interesting philosophicaldiscussion about particles and localized objects in a QFT and will extract herefrom [77, 256, 326, 703, 704]; for QFT we refer to [457, 912, 935, 1015].Many details are omitted and standard QFT techniques are assumed to be knownand we will concentrate here on derivations of KG type equations and the natureof the quantum potential (the Dirac equation will be treated later).

3.1. EMERGENCE OF PARTICLES. The papers [704] are impressivein producing a local operator describing the particle density current for scalar andspinor fields in an arbitrary gravitational and electromagnetic background. Thisenables one to describe particles in a local, general covariant, and gauge invariantmanner. The current depends on the choice of a 2-point Wightman function anda most natural choice based on the Green’s function a la Schwinger- deWitt leadsto local conservation of the current provided that interaction with quantum fieldsis absent. Interactions lead to local nonconservation of current which describeslocal particle production consistent with the usual global description based on theinteraction picture. The material is quite technical but we feel it is importantand will sketch some of the main points; the discussion should provide a goodexercise in field theoretic technique. The notation is indicated as we proceed andwe make no attempt to be consistent with other notation. Thus let gµν be a

classical background metric, g the determinant, and R the curvature. The actionof a Hermitian scalar field φ can be written as

(3.1) S =12

∫d4x|g|1/2[gµν(∂µφ)(∂νφ)−m2φ2 − ξRφ2]

where ξ is a coupling constant. Writing this as S =∫

d4x|g|1/2L the canonicalmommentum vector is πµ = [∂L/∂(∂µφ)] = ∂µφ (standard gµν). The correspond-ing equation of motion is (∇µ∂µ + m2 + ξR)φ = 0 where ∇µ is the covariantderivative. Let Σ be a spacelike Cauchy hypersurface with unit normal vector nµ;the canonical momentum scalar is defined as π = nµπµ and the volume element onΣ is dΣµ = d3x|g(3)|1/2nµ with scalar product (φ1, φ2) = i

∫Σ

dΣµφ∗1

←→∂µφ2 where

a←→∂µb = a∂µg−(∂µa)b. If φi are solutions of the equation of motion then the scalar

product does not depend on Σ. One chooses coordinates (t, x) such that t = c onΣ so that nµ = gµ

0 /√

g00 and the canonical commutation relations become

(3.2) [φ(x), φ(x′)]Σ = [π(x), π(x′)]Σ = 0; [φ(x), π(x′)]Σ = |g(3)|−1/2iδ3(x− x′)

(here x, x′ lie on Σ). This can be written in a manifestly covariant form via

(3.3)∫

Σ

dΣ′µ[φ(x), ∂′

µφ(x′)]χ(x′) =∫

Σ

dΣ′µ[φ(x′), ∂µφ(x)]χ(x′) = iχ(x)

for an arbitrary test function χ. For practical reasons one writes nµ = |g(3)|1/2nµ

where the tilde indicates that it is not a vector. Then ∇µnν = 0 and in factnµ = (|g(3)|1/2/

√g00, 0, 0, 0). It follows that dΣµ = d3xnµ while (2.11) can be

written as n0(x′)[φ(x), ∂′0φ(x′)]Σ = iδ3(x− x′). Consequently

(3.4) [φ(x), π(x′)]Σ = iδ3(x− x′); π = |g(3)|1/2π

Now choose a particular complete orthonormal set of solutions fk(x) of theequation of motion satisfying therefore

(3.5) (fk, fk′) = −(f∗k , f∗

k′) = δkk′ ; (f∗k , fk′) = (fk, f∗

k′) = 0

One can then write φ(x) =∑

k akfk(x) + a†kf∗

k (x) from which we deduce thatak = (fk, φ) and a†

k = −(f∗k , φ) while [ak, a†

k′ ] = δkk′ and [ak, ak′ ] = [a†k, a†

k′ ] =0. The lowering and raising operators ak and a†

k induce the representation ofthe field algebra in the usual manner and ak|0 >= 0. The number operator isN =

∑a†

kak and one defines a two point function W (x, x′) =∑

fk(x)f∗k (x′)

(different definitions appear later). Using the equation of motion one finds thatW is a Wightman function W (x, x′) =< 0|φ(x)φ(x′)|0 > and one has W ∗(x, x′) =W (x′, x). Further, via the equation of motion, for fk, f∗

k one has

(3.6) (∇µ∂µ + m2 + ξR(x))W (x, x′) = 0 = (∇′µ∂′µ + m2 + ξR(x′))W (x, x′)

From the form of W and the commutation relations there results also

(3.7) W (x, x′)|Σ = W (x′, x)|Σ; ∂0∂′0W (x, x′)|Σ = ∂0∂

′0W (x′, x)|Σ;

n0∂′0[W (x, x′)−W (x′, x)]Σ = iδ3(x− x′)


The number operator given by N =∑

a†kak is a global quantity. However a new

way of looking into the concept of particles emerges when ak = (fk, φ), etc. is putinto N; using the scalar product along with the expression for W leads to

(3.8) N =∫

Σ

dΣµ

∫Σ

dΣ′νW (x, s′)

←→∂µ

←→∂′

νφ(x)φ(x′)

By interchanging the names of the coordinates x, x′ and the names of the indicesµ ν this can be written as a sum of two equal terms

(3.9) N =12

∫Σ

dΣµ

∫Σ

dΣ′νW (x, x′)

←→∂µ

←→∂′

νφ(x)φ(x′)+

12

∫Σ

dΣµ

∫Σ

dΣ′νW (x′, x)

←→∂µ

←→∂′

νφ(x′)φ(x)

Using also W ∗(x, x′) = W (x′, x) one sees that (3.9) can be written as N =∫Σ

dΣµjµ(x) where

(3.10) jµ(x) = (1/2)∫

Σ

dΣ′νW (x, x′)

←→∂µ

←→∂′

νφ(x)φ(x′) + h.c.

(where h.c. denotes hermitian conjugate). Evidently the vector jµ(x) should beinterpreted as the local current of particle density. This representation has threeadvantages over N =

∑a†

kak: (i) It avoids the use of ak, a†k related to a particular

choice of modes fk(x). (ii) It is manifestly covariant. (iii) The local current jµ(x)allows one to view the concept of particles in a local manner. If now one puts allthis together with the antisymmetry of

←→∂µ we find

(3.11) jµ = i∑k,k′

f∗k

←→∂µfk′a†

kak′

From this we see that jµ is automatically normally ordered and has the propertyjµ|0 >= 0 (not surprising since N =

∑a†

kak is normally ordered). Further onefinds ∇µjµ = 0 (covariant conservation law) so the background gravitational fielddoes not produce particles provided that a unique vacuum defined by ak|0 >= 0 ex-ists. This also implies global conservation since it provides that N =

∫Σ

dΣµjµ(x)does not depend on time. The extra terms in ∇µjµ = 0 originating from the factthat ∇µ = ∂µ are compensated by the extra terms in N =

∫Σ

dΣµjµ that originatefrom the fact that dΣµ is not written in “flat” coordinates. The choice of vacuumis related to the choice of W (x, x′). Note that although jµ(x) is a local operatorsome nonlocal features of the particle concept still remain because (3.10) involvesan integration over Σ on which x lies. Since φ(x′) satisfies the equation of motionand W (x, x′) satisfies (3.6) this integral does not depend on Σ. However it doesdepend on the choice of W (x, x′). Note also the separation between x and x′ in(3.10) is spacelike which softens the nonlocal features because W (x, x′) decreasesrapidly with spacelike separation - in fact it is negligible when the space like sep-aration is much larger than the Compton wavelength.

We pick this up now in [702]. Thus consider a scalar Hermitian field φ(x) in acurved background satisfying the equation of motion and choose a particular com-plete orthonormal set fk(x) having relations and scalar product as before. The


field φ can be expanded as φ(x) = φ+(x) + φ−(x) where φ+(x) =∑

akfk(x) andφ−(x) =

∑a†

kf∗k (x). Introducing the two point function W+(x, x′) =

∑fk(x)f∗

k (x′)with W−(x, x′) =

∑f∗

k (x)fk(x′) one finds the remarkable result that(3.12)

φ+(x) = i

∫Σ

dΣ′νW+(x, x′)

←→∂ ′

νφ(x′); φ−(x) = −i

∫Σ

dΣ′νW−(x, x′)

←→∂ ′

νφ(x′)

We see that the extraction of φ±(x) from φ(x) is a nonlocal procedure. Notehowever that the integrals in (3.12) do not depend on the choice of the timelikeCauchy hypersurface Σ because W±(x, x′) satisfies the equation of motion withrespect to x′ just as φ(x′) does. However these integrals do depend on the choiceof W±(x, x′), i.e. on the choice of the set fk(x). Now define normal orderingin the usual way, putting φ− on the left, explicitly : φ+φ− := φ−φ+ while theordering of the combinations φ−φ+, φ+φ+, and φ−φ− leaves these combinationsunchanged. Generalize this now by introducing 4 different orderings N(±) andA(±) defined via

(3.13) N+φ+φ− = φ−φ+; N−φ+φ− = −φ−φ+;

A+φ−φ+ = φ+φ−; A−φ−φ+ = −φ+φ−

Thus N+ is normal ordering, N− will be useful, and the antinormal orderings A±can be used via symmetric orderings S+ = (1/2)[N+ + A+] and S− = (1/2)[N− +A−]. When S+ acts on a bilinear combination of fields it acts as the defaultordering, i.e. S+φφ = φφ.

Now the particle current for scalar Hermitian fields can be written as (cf.(3.10))(3.14)

jµ(x) =12

∫Σ

dΣ′ν[W+(x, x′)

←→∂µ←→∂ ′

νφ(x)φ(x; ) + W−(x, x′)←→∂µ←→∂ ′

νφ(x′)φ(x)]

(3.14) can be written in a local form as jµ(x) = (i/2)[φ(x)←→∂µφ+(x)+φ−(x)

←→∂µφ(x)]

(via (3.12)). Using the identities φ+←→∂µφ+ = φ−←→∂µφ− this can be written inthe elegant form jµ = iφ−←→∂µφ+. Similarly using (3.13) this can be written inanother elegant form without explicit use of φ±, namely jµ = (i/2)N−φ

←→∂µφ.

Note that the expression on the right here without the ordering N− vanishesidentically - this peculiar feature may explain why the particle current was notpreviously discovered. The normal ordering N− provides that jµ|0 >= 0 which isrelated to the fact that the total number of particles is N =

∫Σ

dΣµjµ =∑

a†kak.

Alternatively one can choose the symmetric ordering S− and define the particlecurrent as jµ = (i/2)S−φ

←→∂µφ. This leads to the total number of particles N =

(1/2)∑

(a†kak + aka†

k) =∑

[a†kak + (1/2)].

When the gravitational background is time dependent one can introduce a newset of solutions uk(x) for each time t, such that the uk(x) are positive frequencymodes at that time. This leads to functions with an extra time dependence uk(x, t)


that do not satisfy the equation of motion (cf. [704]). Define φ± as in (3.12) butwith the two point functions

(3.15) W+(x, x′) =∑

uk(x, t)u∗k(x′, t′); W−(x, x′) =

∑u∗

k(x, t)uk(x′, t′)

As shown in [704] such a choice leads to a local description of particle creationconsistent with the conventional global description based on the Bogoliubov trans-formation. Putting φ(x) =

∑akfk(x) + a†

kf∗k (x) in (3.12) with (3.15) yields

φ+(x) =∑

Ak(t)uk(x, t) and φ−(x) =∑

A†k(t)u∗

k(x) where

(3.16) Ak(t) =∑

α∗kj(t)aj − β∗

kj(t)a†k; αjk = (fj , uk); βjk(t) = −(f∗

j , uk)

Putting these φ± in jµ = iφ−←−∂ φ+ one finds

(3.17) jµ(x) = i∑k,k′

A†k(t)u∗

k(x, t)←→∂µAk′(t)uk′(x, t)

Note that because of the extra time dependence the fields φ± do not satisy theequation of motion (∇µ∂µ + m2 + ξR)φ = 0 and hence the current (3.17) is notconserved, i.e. ∇µjµ is a nonvanishing local scalar function describing the creationof particles in a local and invariant manner as in [704]. In [702] there follows adiscussion about where and when particles are created with conclusion that thishappens at the spacetime points where the metric is time dependent. Hawkingradiation is then cited as an example. Generally the choice of the 2-point function(3.15) depends on the choice of time coordinate. Therefore in general a naturalchoice of the 2-point function (3.15) does not exist. In [704] an alternative choiceis introduced via W±(x, x′) = G±(x, x′) where G±(x, x′) is determined by theSchwinger- deWitt function. As argued in [704] this choice seems to be the mostnatural since the G± satisfy the equation of motion and hence the particle currentin which φ± are calculated by putting : φ+φ− := φ−φ+ in (3.12) is conserved;this suggests that classical gravitational backgrounds do not create particles (seebelow).

A complex scalar field φ(x) and its Hermitian conjugate φ† in an arbitrarygravitational background can be expanded as

(3.18) φ = φP+ + φA−; φ† = φP− + φA+; φP+ =∑

akfk(x);

φP− =∑

a†kf∗

k ; φA+ =∑

bkfk(x); φA− =∑

b†kf∗k (x)

In a similar manner to the preceeding one finds also

(3.19)

φP+ = i∫

dΣ′νW+(x, x′)

←→∂ ′

νφ(x′); φA+ = i∫Σ

dΣ′νW+(x, x′)

←→∂ ′

νφ†(x′);

φP− = −i

∫Σ


←→∂ ′

νφ†(x′); φA− = −i

∫Σ


←→∂ ′

νφ(x′)


The particle current jPµ (x) and the antiparticle current jA

µ (x) are then (cf. [704])(3.20)

jPµ (x) =

12

∫Σ


←→∂µ←→∂ ′

νφ†(x)φ(x′) + W−(x, x′)←→∂µ←→∂ ′

νφ†(x′)φ(x)];

jAµ (x) =

12

∫Σ


←→∂µ←→∂ ′

νφ(x)φ†(x′) + W−(x, x′)←→∂µ←→∂ ′

νφ(x′)φ†(x)]

Consequently they can be written in a purely local form as

(3.21) jPµ = iφP−←→∂µφP+ + jmix

µ ; jAµ = iφA−←→∂µφA+ − jmix

µ ;

jmixu =

i

2

[φP−←→∂µφA− − φP+←→∂µφA+

]The current of charge j−µ has the form j−µ = jP

µ − jAµ which can be written as (cf.

[704]) j−µ =: iφ†←→∂µφ := i2

[φ†←→∂µφ− φ

←→∂µφ†

]. Using (3.13) this can also be written

as

(3.22) j−µ = N+iφ†←→∂µφ = (i/2)N+[φ†←→∂µφ− φ←→∂µφ†]

The current of total number of particles is now defined as j+µ = jP

µ + jAµ and it is

shown in [704] that j+µ can be written as jµ = j1

µ + j2µ where φ = (1/

√2)(φ1 + iφ2)

(jiµ are two currents of the form (3.14). Therefore using jµ = (i/2)N−φ

←→∂µφ one

can write jµ as j+µ = (i/2)N−[φ1

←→∂µφ1 + φ2

←→∂µφ2]. Finally one shows that this can

be written in a form analogous to (3.22) as j+µ = (i/2)N−[φ†←→∂µφ + φ

←→∂µφ†]. This

can be summarized by defining currents q±µ = (1/2)[φ†←→∂µφ ± φ←→∂µφ†] leading to

j±µ = Nµq±µ . The current q+µ vanishes but N−q+

µ does not vanish. These resultscan be easily generalized to the case where the field interacts with a backgoundEM field (as in [704]). The equations are essentially the same but the derivatives∂µ are replaced by the corresponding gauge covariant derivatives and the particle2-point functions WP± are not equal to the antiparticle 2-point functions WA±.As in the gravitational case in the case of interaction with an EM background threedifferent choices for the 2-point functions exist and we refer to [704] for details.

REMARK 2.3.1 In a classical field theory the energy-momentum tensor(EMT) of a real scalar field is

(3.23) Tµν = (∂µφ)(∂νφ)− (1/2)gµν [gαβ(∂αφ)(∂βφ)−m2φ2]

Contrary to the conventional idea of particles in QFT the EMT is a local quantity.Therefore the relation between the definition of particles and that of EMT is notclear in the conventional approach to QFT in curved spacetime. Here one canexploit the local and covariant description of particles to find a clear relationbetween particles and EMT. One has to choose some ordering of the operators in(3.23) just as a choice of ordering is needed in order to define the particle current.Although the choice is not obvious it seems natural that the choice for one quantityshould determine the choice for the other. Thus if the quantum EMT is definedvia : Tµν := N+Tµν then the particle current should be defined as N−iφ

←→∂µφ.

The nonlocalities related to the extraction of φ+ and φ− from φ needed for the


definitions of the normal orderings N+ and N− appear both in the EMT and in theparticle current. Similarly if W± is chosen as in W+(x, x′) =

∑fk(x)f∗

k (x′) forone quantity then it should be chosen in the same way for the other. The choicesas above lead to a consistent picture in which both the energy and the number ofparticles vanish in the vacuum |0 > defined by ak|0 >= 0. Alternatively if W± ischosen as in (3.15) for the definition of particles it should be chosen in the same wayfor the definition of the EMT. Assume for simplicity that spacetime is flat at somelate time t. Then the normally ordered operator of the total number of particlesat t is N(t) =

∑q A†

q(t)Aq(t) (cf. (3.16)) while the normally ordered operator ofenergy is H(t) =

∑q ωqA

†(t)Aq(t) (note here q ∼ q). Owing to the extra timedependence it is clear that both the particle current and the EMT are not conservedin this case. Thus it is clear that the produced energy exactly corresponds tothe produced particles. A similar analysis can be caried out for the particle-antiparticle pair creation caused by a classical EM background. Since the energyshould be conserved this suggests that W± should not be chosen as in (3.15),i.e. that classical backgrounds do not cause particle creation (see [702] for morediscussion). The main point in all this is that particle currents as developed abovecan be written in a purely local form. The nonlocalities are hidden in the extractionof φ± from φ. The formalism also reveals a relation between EM and particlessuggesting that it might not be consistent to use semiclassical methods to describeparticle creation; it also suggests that the vacuum energy might contribute to darkmatter that does not form structures, instead of contributing to the cosmologicalconstant.

3.2. BOSONIC BOHMIAN THEORY. We follow here [703] concern-ing Bohmian particle trajectories in relativistic bosonic and fermionic QFT. Firstwe recall that there is no objection to a Bohmian type theory for QFT and nocontradiction to Bell’s theorems etc. (see e.g. [77, 126, 256, 326]). Withoutdiscussing all the objections to such a theory we simply construct one followingNikolic (cf. also [180, 453, 588] for related information). Thus consider firstparticle trajectories in relativistic QM and posit a real scalar field φ(x) satisfy-ing the Klein-Gordon equation in a Minkowski metric ηµν = diag(1,−1,−1,−1)written as (∂2

0 − ∇2 + m2)φ = 0. Let ψ = φ+ with ψ∗ = φ− correspondto positive and negative frequency parts of φ = φ+ + φ−. The particle cur-rent is jµ = iψ∗←→∂µψ and N =

∫d3xj0 is the positive definite number of parti-

cles (not the charge). This is most easily seen from the plane wave expansionφ+(x) =

∫d3ka(κ)exp(−ikx)/

√(2π)32k0 since then N =

∫d3ka†(κ)a(κ) (see

above and [702, 704] where it is shown that the particle current and the decom-position φ = φ+ + φ− make sense even when a background gravitational field orsome other potential is present). One can write also j0 = i(φ−π+ − φ+π−) whereπ = π+ + π− is the canonical momentum (cf. [471]). Alternatively φ may beinterpreted not as a field containg an arbitrary number of particles but rather as aone particle wave function. Here we note that contrary to a field a wave functionis not an observable and so doing we normalize φ here so that N = 1. The currentjµ is conserved via ∂µjµ = 0 which implies that N =

∫d3xj0 is also conserved,

i.e. dN/dt = 0. In the causal interpretation one postulates that the particle has

the trajectory determined by dxµ/dτ = jµ/2mψ∗ψ. The affine parameter τ canbe eliminated by writing the trajectory equation as dx/dt = j(t,x)/j0(t,x) wheret = x0, x = (x1, x2, x3) and j = (j1, j2, j3). By writing ψ = Rexp(iS) where R, S)are real one arrives at a Hamilton-Jacobi (HJ) form dxµ/dτ = −(1/m)∂muS andthe KG equation is equivalent to

(3.24) ∂µ(R2∂µS) = 0;(∂µS)(∂µS)

2m− m

2+ Q = 0

Here Q = −(1/2m)(∂µ∂µR/R is the quantum potential. One has put here c = = 1 and reinserted we would have

(3.25)(∂µS)(∂µS)

2m− c2m

2− 2

2m

∂µ∂µR

R= 0

From the HJ form and (3.24) plus the identity d/dτ = (dxµ/dt)∂µ one arrivesat the equations of motion m(d2xµ/dτ2) = ∂µQ. A typical trajectory arisingfrom dx/dt = j/j0 could be imagined as an S shaped curve in the t − x plane(with t horizontal) and cut with a vertical line through the middle of the S. Thevelocity may be superluminal and may move backwards in time (at points wherej0 < 0). There is no paradox with backwards in time motion since it is physicallyindistinguishable from a motion forwards with negative energy. One introduces aphysical number of particles via Nphys =

∫d3x|j0|. Contrary to N =

∫d3xj0 the

physical number of particles is not conserved. A pair of particles one with positiveand the other with negative energy may be created or annihilated; this resemblesthe behavior of virtual particles in convential QFT.

Now go to relativistic QFT where in the Heisenberg picture the Hermitianfield operator φ(x) satisfies

(3.26) (∂20 −∇2 + m2)φ = J(φ)

where J is a nonlinear function describing the interaction. In the Schrodingerpicture the time evolution is determined via the Schrodinger equation (SE) in theform H[φ,−iδ/δφ]Ψ[φ, t] = i∂tΨ[φ, t] where Ψ is a functional with respect to φ(x)and a function of t. A normalized solution of this can be expanded as Ψ[φ, t] =∑∞

−∞ Ψn[φ, t] where the Ψn are unnormalized n-particle wave functionals. Sinceany (reasonable) φ(x) can be Fourier expanded one can write

(3.27) Ψn[φ, t] =∫

d3k1 · · · d3kncn(k(n), t)Ψn,k(n) [φ]

where k(n) = k1, · · · ,kn. These functionals in (3.27) constitute a completeorthonormal basis which generalizes the basis of Hermite functions and they satisfy

(3.28)∫DφΨ∗

0[φ]φ(x1) · · ·φ(xn′)Ψn,k(n) [φ] = 0 (n = n)

For free fields (i.e. when J = 0 in (3.26) one has

(3.29) cn(k(n), t) = cn(k(n))e−iωn(k(n))t; ωn = E0 +n∑1

√k2

j + m2

where E0 is the vacuum energy. In this case the quantities |cn(k(n), t)|2 do notdepend on time so the number of particles (corresponding to the quantized versionof N =

∫d3xj0) is conserved. In a more general situation with interactions the

SE leads to a more complicated time depenedence of the coefficients cn and thenumber of particles is not conserved. Now the n-particle wave function is

(3.30) ψn(x(n), t) =< 0|φ(t,x1) · · · φ(t,xn)|Ψ >

(the multiplication of the right side by (n!)−1/2 would lead to a normalized wavefunction only if Ψ = Ψn). The generalization of (3.30) to the interacting case isnot trivial because with an unstable vacuum it is not clear what is the analogueof < 0|. Here the Schrodinger picture is more convenient where (3.30) becomes

(3.31) ψn(x(n), t) =∫DφΨ∗

0[φ]exp(−iφ0(t))φ(x1) · · ·φ(xn)Ψ[φ, t]

where φ0(t) = −E0t. For the interacting case one uses a different phase φ0(t)defined via an expansion, namely

(3.32) U(t)Ψ0[φ] = r0(t)exp(iφ0(t))Ψ0[φ] +∞∑1

· · ·

where r0(t) ≥ 0 and U(t) = U(φ,−iδ/δφ, t] is the unitary time evolution operator.One sees that even in the interacting case only the Ψn part of Ψ contributesto (3.31) so Ψn can be called the n-particle wave functional. The wave function(3.30) can also be generalized to a nonequaltime wave function ψn(x(n)) = Sxj <

0|φ(x1) · · · φ(xn)|Ψ > (here Sxj denotes symmetrization over all xj which isneeded because the field operators do not commute for nonequal times. For theinteracting case the nonequaltime wave function is defined as a generalization of(3.30) with the replacements

(3.33) φ(xj) → U†(tj)φ(xj)U(tj); Ψ[φ, t] → U†(t)Ψ[φ, t] = Ψ[φ];

e−iφ0(t) → e−iφ0(t1)U(t1)

followed by symmetrization.

In the deBroglie-Bohm (dBB) interpretation the field φ(x) has a causal evo-lution determined by

(3.34) (∂20 −∇2 + m2)φ(x) = J(φ(x))−

(δQ[φ, t]δφ(x)

)φ(x)=φ(x)

;

Q = − 12|Ψ|

∫d3x

δ2|Ψ|δφ2(x)

where Q is the quantum potential again. However the n particles attributed tothe wave function ψn also have causal trajectories determined by a generalizationof dx/dt = j/j0 as

(3.35)dxn,j

dt=

(ψ∗

n(x(n))←→∇jψn(x(n))

ψ∗n(x(n))

←→∂tj

ψn(x(n))

)t1=···=tn=t


These n-particles have well defined trajectories even when the probability (in theconventional interpretation of QFT) of the experimental detection is equal to zero.In the dBB interpretation of QFT we can introduce a new causally evolving “ef-fectivity” parameter en[φ, t] defined as

(3.36) en[φ, t] = |Ψn[φ, t]|2/∞∑n′|Ψn′ [φ, t]|2

The evolution of this parameter is determined by the evolution of φ given via (3.34)and by the solution Ψ =

∑Ψ of the SE. This parameter might be interpreted as

a probability that there are n particles in the system at time t if the field is equal(but not measured!) to be φ(x) at that time. However in the dBB theory onedoes not want a stochastic interpretation. Hence assume that en is an actualproperty of the particles guided by the wave function ψn and call it theeffectivity of these n particles. This is a nonlocal hidden variable attributedto the particles and it is introduced to provide a deterministic description of thecreation and destruction of particles. One postulates that the effective mass of aparticles guided by ψn is meff = enm and similarly for the energy, momentum,charge, etc. This is achieved by postulating that the mass density is ρmass(x, t) =m∑∞

1 en

∑n1 δ3(x − xn,j(t)) and similarly for other quantities. Thus if en = 0

such particles are ineffective, i.e. their effect is the same as if they didn’t existwhile if en = 1 they exist in the usual sense. However the trajectories are definedeven for the particles for which en = 0 and QFT is a theory of an infinite numberof particles although some of them may be ineffective (conventionally one wouldsay they are virtual). We will say more about this later.

3.3. FERMIONIC THEORY. This extraction from [701] (cf. also [325])becomes even more technical but a sketch should be rewarding; there is moredetail and discussion in [701]. The Dirac equation in Minkowski space ηµν =diag(1,−1,−1,−1) is iγµ∂µ −m)ψ(x) = 0 where x = (xi) = (t,x) with x ∈ R3

(cf. Section 2.1.1). A general solution can be written as ψ(x) = ψP (x) + ψA(x)where the particle and antiparticle parts can be expanded as ψP =

∑bkuk(x) and

ψA =∑

d∗kvk(x). Here uk (resp. vk) are positive (resp. negative) frequency 4-spinors that, together, form a complete orthonormal set of solutions to the Diracequation. The label k means (k, s) where s = ±1/2 is the spin label. WritingΩP (x, x′) =

∑uk(x)u†

k(x′) and ΩA(x, x′) =∑

vk(x)v†k(x′) one can write

(3.37) ψP =∫

d3x′ΩP (x, x′)ψ(x′); ψA(x) =∫

d3x′ΩA(x, x′)ψ(x′)

where t = t′. The particle and antiparticle currents are jPµ = ψP γµψP and jA

µ =ψAγµψA where ψ = ψ†γ0. Since ψP and ψA satisfy the Dirac equation the currentsjPµ , jA

µ are separately conserved, i.e. ∂µjPµ = ∂µjA

µ = 0. One postulates thentrajectories of the form

(3.38)dxP

dt=

jP (t,xP )jP0 (t,x)

;dxA

dt=

jA(t,xA)jA0 (t,xA)

where j = (j1, j2, j3) for a causal interpretation of the Dirac equation. Now inQFT the coefficients bk and d∗k become anticommuting operators with b†k andd†k creating particles and antiparticles while bk and dk annihilate them. In theSchrodinger picture the field opperators ψ(x) and ψ†(x) satisfy the commutationrelations ψa(x), ψ†

a′(x′) = δaa′δ3(x − x′) while other commutators vanish (a isthe spinor index). These relations can be represented via

(3.39) ψa(x) =1√2

[ηa(x) +

δ

δη∗a(x)

]; ψ†

a(x) =1√2

[η∗

a(x) +δ

δηz(x)

]where ηa, η∗

a are anticommuting Grassmann numbers satisfying ηa(x), ηa′(x′) =η∗

a(x), η∗a′(x′) = ηa(x), η∗

a′(x′) = 0. Next introduce a complete orthonormalset of spinors uk(x) and vk(x) which are equal to the spinors uk(x) and vk(x) att = 0. An arbitrary quantum state may then be obtained by acting with creationoperators

(3.40) b†k =∫

d3xψ†(x)uk(x); d†k =∫

d3xv†k(x)ψ(x)

on the vacuum |0 >= |Ψ0 > represented by

(3.41) Ψ0[η, η†] = Nexp∫

d3x

∫d3x′η†(x)Ω(x,x′)η(x′)

Here Ω(x,x′) = (ΩA − ΩP )(x,x′), N is a constant such that < Ψ0|Ψ0 >= 1 andthe scalar product is < Ψ|Ψ >=

∫D2ηΨ∗[η, η†Ψ′[η, η†]; also D2 = DηDη† and

Ψ∗ is dual (not simply the complex conjugate) to Ψ. The vacuum is chosen suchthat bkΨ0 = dkΨ0 = 0. A functional Ψ[η, η†] can be expanded as Ψ[η, η†] =∑

cKΨK [η, η†] where the set ψK is a complete orthonormal set of Grassmannvalued functionals. This is chosen so that each ΨK is proportional to a functionalof the form b†k1

· · · b†knPd†k′

1· · · d†k′

nA

Ψ0 which means that each ΨK has a definite

number nP of particles and nA of antiparticles. Therefore one can write Ψ[η, η†] =∑∞nP ,nA=0 ΨnP ,nA

[η, η†] where the tilde denotes that these functionals, in contrastto Ψ and ΨK , do not have unit norm. Time dependent states Ψ[η, η†, t] can beexpanded as

(3.42) Ψ[η, η†, t] =∑K

cK(t)ΨK [η, η†] =∞∑

np,nA=0

ΨnP ,nA[η, η†, t]

The time dependence of the c-number coefficients cK(t) is governed by the func-tional SE

(3.43) H[ψ, ψ†]Ψ[η, η†, t] = i∂tΨ[η, η†, t]

Since the Hamiltonian H is a Hermitian operator the norms < Ψ(t)|Ψ(t) >=∑|cK(t)|2 do not depend on time. In particular if H is the free Hamiltonian

(i.e. the Hamiltonian that generates the second quantized free Dirac equation)then the quantities |cK(t)| do not depend on time, which means that the averagenumber of particles and antiparticles does not change with time when there areno interactions.

Next introduce the wave function of nP particles and nA antiparticles via

(3.44) ψnP ,nA≡ ψb1···bnP

d1···dnA(x1, · · · ,xnP

,y1 · · · ,ynA, t)

It has nP + nA spinor indices and for free fields the (unnormalized) wave functioncan be calculated using the Heisenberg picture as

(3.45) ψnP ,nA=< 0|ψP

b1(t,x1) · · · ψA†dnA

(t,ynA)|Ψ >

where ψP and ψA are extracted from ψ using (3.37). In the general interactingcase the wave function can be calculated using the Schrodinger picture as

(3.46) ψnP ,nA=∫D2ηΨ∗

0[η, η†]e−iφ0(t)ψPb1(x1) · · · ψA†

dnA(ynA

)Ψ[η, η†, t]

Here the phase φ0(t) is defined by an expansion as in (3.42), namely

(3.47) U(t)Ψ0[η, η†] = r0(t)exp(iφ0(t))Ψ0[η, η†] +∑

(nP ,nA)=(0,0)

· · ·

where r0(t) ≥ 0 and U(t) = U [ψ, ψ†, t] is the unitary time evolution operator thatsatisfies the SE (3.43). The current attributed to the ith corpuscle (particle orantiparticle) in the wave function ψnP ,nA

is jµ(i) = ψnP ,nAγµ(i)ψnP ,nA

where onewrites

(3.48) ψΓiψ = ψa1···ai···an(Γ)aia′

iψa1···a′

i···an;

ψa1···an= ψ∗

a′1···a′

n(γ0)a′

1a1 · · · (γ0)a′nan

Hence the trajectory of the ith corpuscle guided by the wave function ψnP ,nAis

given by the generalization of (3.38), namely dxi/dt = ji/j0(i).

We now need a causal interpretation of the processes of creation and destruc-tion of particles and antiparticles. For bosonic fields this was achieved by intro-ducing the effectivity parameter in Section 1.3.2 but this cannot be done for theGrassmann fields η, η† because Ψ∗[η, η†, t]Ψ[η, η†, t] is Grassmann valued and can-not be interpreted as a probability density. Hence another formulation of fermionicstates is developed here, more similar to the bosonic states. First the notion of thescalar product can be generalized in such a way that it may be Grassmann valuedwhich allows one to write Ψ[η, η†, t] =< η, η†|Ψ(t) > and 1 =

∫D2η|η, η† >< η, η†|

(cf. [457]). We can also introduce

(3.49) < φ, φ†|η, η† >=∑K

< φ, φ†|ΨK >< ΨK |η, η† >=∑K

ΨK [φ, φ†]Ψ∗K [η, η†]

so one sees that the sets ΨK [η, η†] and ΨK [φ, φ†] are two representations of thesame orthonormal basis |ΨK > for the same Hilbert space of fermionic states.In other words the state |Ψ(t) > can be represented as Ψ[φ, φ†, t] =< φ, φ†|Ψ(t) >which can be expanded as

(3.50) Ψ[φ, φ†, t] =∑K

cK(t)ΨK [φ, φ†] =∞∑

nP ,nA=0

ψnP ,nA[φ, φ†, t]

Putting the unit operator 1 =∫D2φ|φ, φ† >< φ, φ†| in the expression for <

Ψ(t),Ψ(t) > we see that the time independent norm can be written as

(3.51) < Ψ(t)|Ψ(t) >=∫D2φΨ∗[φ, φ†, t)]Ψ[φ, φ†, t]

Therefore the quantity ρ[φ, φ†, t] = Ψ∗[φ, φ†, t]Ψ[φ, φ†, t] can be interpreted as apositive definite probability density for spinors φ, φ† to have space dependenceφ(x) and φ†(x) respectively at time t. The SE (3.43) can also be written in theφ-representation as HφΨ[φ, φ†, t] = i∂tΨ[φ, φ†, t] where the Hamiltonian Hφ isdefined by its action on wave functionals Ψ[φ, φ†, t] determined via(3.52)

Hφ[φ, φ†, t] =∫D2η

∫D2φ′ < φ, φ†|η, η† > H < η, η†|φ′, φ

′† > Ψ[φ′, φ′†, t]

where H = H[ψ, ψ†] is the Hamiltonian of (3.43).

One can now obtain a causal interpretation of a quantum system describedby a c-number valued wave function satisfying a SE. The material is written foran n-dimensional vector φ but in a form that generalizes to infinite dimensions.The wave function ψ(φ, t) satisfies the SE Hψ = i∂tψ where H is an arbitraryHermitian Hamiltonian written in the φ representation. The quantity ρ = ψ∗ψ isthe probability density for the variables φ and the average velocity is

(3.53) d < φ > (t)/dt =∫

dnφρ(φ, t)u(φ, t); u = iψ∗[H, φ]ψ/ψ∗ψ

Introduce a source J via J = (∂ρ/∂t)+∇(ρu) (note e.g. for the example of (3.52) Jdoes not vanish even though it frequently will vanish). One wants to find a quantityv(φ, t) that has the property (3.53) in the form d < φ > (t)/dt =

∫dnφρ(φ, t)v(φ, t)

but at the same time satisfies the equivariance property ∂tρ+∇(ρv) = 0. These twoproperties allow one to postulate a consistent causal interpretation of QM in whichφ has definite values at each time t determined via dφ/dt = v(φ, t). In particularthe equivariance provides that the statistical distribution of the variables φ is givenby ρ for any time t provided that it is given by ρ for some initial time t0. WhenJ = 0 then v = u which corresponds to the dBB interpretation. The aim now is togeneralize this to the general case of v in the form v = u+ρ−1E where E(φ, t) is thequantity to be determined. From ∂tρ+ ∇(ρv) = 0 we see that E must be a solutionof the equation ∇E = −J . Now let E be some particular solution of this equation;then E(φ, t) = e(t) + E(φ, t) is also a solution for an arbitrary φ independentfunction e(t). Comparing with (3.53) one sees that

∫dnφE = 0 is required. This

fixes the function e to be e(t) = −V −1∫

dnφE(φ, t) where V =∫

dnφ. Thus itremains to choose E and in [703] one takes E such that E = 0 when J = 0so that E = 0 when J = 0 as well; thus v = u when J = 0. There is stillsome arbitrariness in E so take E = ∇Φ where ∇2Φ = −J , which is solved viaΦ(φ, t)

∫dnφ′G(φ, φ′)J(φ′, t), so that ∇2G(φ, φ′) = −δn(φ− φ′). The solution can

be expressed as a Fourier transform G(φ, φ′) =∫

(dnk/(2π)n)exp[ik(φ − φ′)]/k2.


To eleminate the factor 1/(2π)n one uses a new integration variable χ = k/2π andwe obtain

(3.54) Φ(φ, t) =∫

dnχ

∫dnφ′[exp(2iπχ(φ− φ′)]J(φ′, t)/(2π)2χ2]

Now for a causal interpretation of fermionic QFT one writes first for simplicityA[x] for functionals of the form A[φ, φ†, t,x] and introduces

(3.55) ua[x] = iΨ∗[Hφ, φa(x)]Ψ

Ψ∗Ψ; u∗

a[x] = iΨ∗[Hφ, φ∗

a(x)]ΨΨ∗Ψ

where Ψ = Ψ[φ, φ†, t]. Next introduce the source

(3.56) J =∂ρ

∂t+∑

a

∫d3x

[δ(ρua[x])δφz(x)

+δ(ρu∗

a[x])δφ∗

a(x)

]where ρ = Ψ∗Ψ. Introduce now the notation α · β =

∑a

∫d3x[αa(x)βa(x) +

α∗a(x)β∗

a(x)] and (3.51) generalizes to

(3.57) Φ[φ, φ†, t] =∫D2χ

∫D2φ′ e

2πiχ·(φ−φ′)

(2π)2χ · χ J [φ′, φ′†, t]

Then write for V =∫D2φ

(3.58) Ea[x] =δΦ

δφa(x); E∗

a [x] =δΦ

δφ∗a(x)

;

ea(t,x) = −V −1

∫D2φEa[φ, φ†, t,x]; e∗a(t,x) = −V −1

∫D2φE∗

a[φ, φ†, t,x]

The corresponding velocities are then

(3.59)va[x] = ua[x] + ρ−1(ea(t,x) + Ea[x]); v∗

a[x] = u∗a[x] + ρ−1(e∗a(t,x) + E∗

a [x])

Next introduce hidden variables φ(t,x) and φ†(t,x) with causal evolution giventhen by

(3.60)∂φa(t,x)

∂t= va[φ, φ†, t,x];

∂φ∗a(t,x)∂t

= v∗a[φ, φ†, t,x]

where it is understood that the right sides are calculated at φ(x) = φ(t,x) etc. Inanalogy with the bosonic fields treated earlier one introduces effectivity parametersguided by the wave function ψnP ,nA

given by

(3.61) enP ,nA[φ, φ†, t] =

|ΨnP ,nA[φ, φ†, t]|2∑

n′P ,n′

A|Ψn′

P ,n′A[φ, φ†, t]|2

REMARK 2.3.2. Concerning the nature of the effectivity parameter weextract from [701] as follows. In the bosonic theory the analogue of (3.61) is

(3.62) en[φ, t] =|Ψn[φ], t|2∑n′ |Ψn′ [φ], t|2

φ = φ1, · · · , φNs where Ns is the number of different particle species. Now

the measured effectivity can be any number between 0 and 1 and this is no con-tradiction since if different Ψn in the expansion do not overlap in the φ space


then they represent a set of nonoverlapping “channels” for the causally evolvingfield φ. The field necessarily enters one and only one of the channels and one seesthat en = 1 for the nonempty channel with en′ = 0 for all empty channels. Theeffect is the same as if the wave functional Ψ “collapsed” into one of the statesψn with a definite number of particles. In a more general situation different Ψn ofthe measured particles may overlap. However the general theory of ideal quantummeasurements (cf. [126]) provides that the total wave functional can be writtenagain as a sum of nonoverlappiing wave functionals in the φ space, where oneof the fields represents the measured field, while the others represent fields of themeasuring apparatus. Thus only one of the Ψ in (3.62) becomes nonempty withthe corresponding en = 1 while all the other en′ = 0. The essential point is thatfrom the point of view of an observer who does not know the actual field configura-tion the probability for such an effective collapse of the wave functional is exactlyequal to the usual quantum mechanical probability for such a collapse. Hence thetheory has the same statistical properties as the usual theory. In the case whenall the effectivities are less than 1 (i.e. the wave functional has not collapsed) thetheory does not agree nor disagree with standard theory; effectivity is a hiddenvariable. This agrees with the Bohmian particle positions which agree with thestandard quantum theory only when the wave function effectively collapses into astate with a definite particle position. Similar comments apply to the fermionicpicture. In an ideal experiment in which the number of particles is measured, dif-ferent ΨnP ,nA

do not overlap in the (φ, φ†) space and the fields φ, φ† necessarilyenter into a unique “channel” ΨnP ,nA

, etc.

REMARK 2.3.3. In [711] one addresses the question of statistical trans-parency. Thus the probabilitistic interpretation of the nonrelativistic SE does notwork for the relativistic KG equation (∂µ∂µ + m2)ψ = 0 (where x = (x, t) and = c = 1) since |ψ|2 does not correspond to a probability density. There is aconserved current jµ = iψ∗←→∂µψ (where a

←→∂µb = a∂µb− b∂µa) but the time com-

ponent j0 is not positive definite. In [701, 703] the equations that determine theBohmain trajectories of relativistic quantum particles described by many particlewave functions were written in a form requiring a preferred time coordinate. How-ever a preferred Lorentz frame is not necessare (cf. [105]) and this is developedin [711] following [105, 703]. First note that as in [105, 703] it appears thatparticles may be superluminal and the principle of Lorentz covariance does notforbid superluminal velocities and conversly superluminal velocities do not leadto causal paradoxes (cf. [105, 711]). As noted in [105] the Lorentz-covariantBohmian interprtation of the many particle KG equation is not statistically trans-parent. This means that the statistical distribution of particle positions cannot becalculated in a simple way from the wave function alone without the knowledgeof particle trajetories. One knows that classcal QM is statistically transparent ofcourse and this perhaps helps to explain why Bohmian mechanics has not attractedmore attention. However statistical transparency (ST) may not be a fundamentalproperty of nature as the following facts suggest:

• Classical mechanics, relativistic or nonrelativistic, is not ST.

• Relativistic QM based on the KG equation (or some of its generalizations)is not ST.

• The relativistic Dirac equation is ST but its many particle relativisticgeneralization is not (unless a preferred time coordinate is determined inan as yet unknown dynamical manner).

• Nonrelativistic QM is ST but not completely so since it distinguishes thetime variable (e.g. ρ(x1, x2, t) is not a probability density).

• The background independent quantum gravity based on the Wheeler-DeWitt (WDW) equation lacks the notion of time and is not ST.

The upshot is that since statistical probabilities can be calculated via Bohmiantrajectories that theory is more powerful than other interpretations of general QM(see [711] for discussion on this). Now let φ(x) be a scalar field operator satisfyingthe KG equation (an Hermitian uncharged field for simplicity so that negativevalues of the time component of the current cannot be interpreted as negativelycharged particles). The corresponding n-particle wave function is (cf. [703])

(3.63) ψ(x1, · · · , xn) = (n!)−1/2Sxa < 0|φ(x1) · · · φ(xn)|n >

Here Sxa (a = 1, · · · , n) denotes the symmetrization over all xa which is neededbecause the field operators do not commute for nonequal times. The wave functionψ satisfies n KG equations

(3.64) (∂µa ∂aµ + m2)ψ(x1, · · · , xn) = 0

Although the operator φ is Hermitian the nondiagonal matrix element ψ definedby (3.63) is complex and one can introduce n real 4-currrents jµ

a = iψ∗←→∂µaψ each

of which is separately conserved via ∂µa jaµ = 0. Equation (3.64) also implies

(3.65)

(∑a

∂µa ∂aµ + nm2

)ψ(x1, · · · , xn) = 0

and the separate conservation equations imply that∑

a ∂µa jaµ = 0. Now write

ψ = Rexp(iS) with R and S real. Then (3.65) is equivalent to a set of twoequations(3.66)∑

a

∂µa (2∂aµS) = 0; −

∑a(∂µ

a S)(∂aµS)2m

+nm

2+ Q = 0; Q =

12m

∑a ∂µ

a ∂aµR

2mR

where Q is the quantum potential. The first equation is equivalent to a currentconservation equation while the second is the quantum analogue of the relativisticHJ equation for n particles. The Bohmian interpretation consistists in postulat-ing the existence of particle trajectories xµ

a(s) satisfying dxµa/ds = −(1/m)∂µ

a swhere s is an affine parameter along the n curves in the 4-dimensional Minkowskispace. This equation has a form identical to the corresponding classical rela-tivistic equation and can also be written as dxµ

a/ds = jµa /2mψ∗ψ. Hence using

d/ds =∑

a(dxµz /ds)∂aµ one finds the equations of motion

(3.67) md2xµ

a

ds2= ∂µ

a Q

71

Note that the equations above for the particle trajectories are nonlocal but stillLorentz covariant. The Lorentz covariance is a consequence of the fact that thetrajectories in spacetime do not depend on the choice of affine parameter s (cf.[105]). Instead, by choosing n “initial” spacetime positions xa, the n trajectoriesare uniquely determined by the vector fields jµ

a or −∂µa S (i.e. the trajectories are

integral curves of these vector fields). The nonlocality is encoded in the fact thatthe right hand side of (3.67) depends not only on xa but also on all the otherxa′ . This is a consequence of the fact that Q(x1, · · · , xn) in (3.66) is not of theform

∑a Qa(x1, · · · , xn), which in turn is related to the fact that S(x1, · · · , xn)

is not of the form∑

a S(xa). Note also that the fact that we parametrize alltrajectories with the same parameter s is not directly related to the nonlocality,because such a parametrization can be used even in local classical physics. Whenthe interactions are local then one can even use another parameter sa for eachcurve but when the interactions are not local one must use a single parameter s;new separate parameters could only be used after the equations are solved. In thenonrelativistic limit all wave function frequencies are (approximately) equal to m sofrom jµ

a ψ∗←→∂µaψ all time components are equal and given by j0

a = 2mψ∗ψ = ρ whichdoes not depend on a. Writing then ρ(x1, · · · ,xn) = ρ(x1, · · · , xn)|t1=···=tn=t

one obtains ∂tρ +∑

a ∂iajai = 0 and this implies that ρ can be interpreted as a

probability density. In the full relativistic there is generally no analogue of such afunction ρ. We refer to [711] for more discussion.

4. DeDONDER, WEYL, AND BOHM

We go here to a fascinating paper [708] which gives a manifestly covariantcanonical method of field quantization based on the classical DeDonder-Weyl (DW)formulation of field theory (cf. also Appendix A for some background on DWtheory following [586]). The Bohmian formulation is not postulated for intepre-tational purposes here but derived from purely technical requirements, namelycovariance and consistency with standard QM. It arises automatically as a part ofthe formalism without which the theory cannot be formulated consistently. Thistogether with the results of [701, 711] suggest that it is Bohmian mechanics thatmight be the missing bridge between QM and relativity; further (as will be seenlater) it should play an important role in cosmology. The classical covariant canon-ical DeDonder-Weyl formalism is given first following [586] and for simplicity onereal scalar field in Minkowski spacetime is used. Thus let φ(x) be a real scalarfield described by

(4.1) A =∫

d4xL; L =12(∂µφ)(∂µφ)− V (φ)

As usual one has

(4.2) πµ =∂L

∂(∂µφ)= ∂µφ; ∂µφ =

∂H

∂πµ; ∂µπµ = −∂H

∂φ

where the scalar DeDonder-Weyl (DDW) Hamilonian (not related to the energydensity) is given by the Legendre transform H(πµ, φ) = πµ∂µφ−L = (1/2)πµπµ +V . The equations (4.2) are equivalent to the standard Euler-Lagrange (EL) equa-tions and by introducing the local vector Sµ(φ(x), x) the dynamics can also be

4. WEYL, AND BOHMD DONDER,e


described by the covariant DDW HJ equation and equations of motion

(4.3) H

(∂Sα

∂φ, φ

)+ ∂µSµ = 0; ∂µφ = πµ =

∂Sµ

∂φ

Note here ∂µ is the partial derivative acting only on the second argument ofSµ(φ(x), x); the corresonding total derivative is dµ = ∂µ + (∂µφ)(∂/∂φ). Notethat the first equation in (4.3) is a single equation for four quantities Sµ so thereis a lot of freedom in finding solutions. Nevertheless the theory is equivalent toother formulations of classical field theory. Now following [533] one considers therelation between the covariant HJ equation and the conventional HJ equation; thelatter can be derived from the former as follows. Using (4.2), (4.3) takes the form(1/2)∂φSµ∂φSµ +V +∂µSµ = 0. Then using the equation of motion in (4.3) writethe first term as

(4.4)12

∂Sµ

∂φ

∂Sµ

∂φ=

12

∂S0

∂φ

∂S0

∂φ+

12(∂iφ)(∂iφ)

Similarly using (4.3) the last term is ∂µSµ = ∂0S0 +diS

i− (∂iφ)(∂iφ). Now intro-duce the quantity S =

∫d3xS0 so [∂S0(φ(x), x)/∂φ(x)] = [δS([φ(x, t)], t)/δφ(x, t)]

where δ/δφ(x, t) ≡ [δ/δφ(x)]φ(x)=φ(x,t) is the space functional derivative. Puttingthis together gives then

(4.5)∫

d3x

[12

(δS

δφ(x, t)

)2

+12(∇φ)2 + V (φ)

]+ ∂tS = 0

which is the standard noncovariant HJ equation. The time evolution of φ(x, t) isgiven by ∂tφ(x, t) = δS/δφ(x, t) which arises from the time component of (4.3).Note that in deriving (4.5) it was necessary to use the space part of the equationsof motion (4.3) (this does not play an important role in classical physics but isimportant here). Now for the Bohmian formulation look at the SE HΨ = i∂tΨwhere we write

(4.6) H =∫

d3x

[−2

2

(δ

δφ(x)

)2

+12(∇φ)2 + V (φ)

];

Ψ([φ(x)], t) = R([φ(x)], t)eiS(([φ(x)],t)/

Then the complex SE equation is equivalent to two real equations

(4.7)∫

d3x

[12

(δS

δφ(x)

)2

+12(∇φ)2 + V (φ) + Q

]+ ∂tS = 0;

∫d3x

[δR

δφ(x)δS

δφ(x)+ J

]+ ∂tR = 0; Q = − 2

2R

δ2R

δφ2(x); J =

R

2δ2S

δφ2(x)The second equation is also equivalent to

(4.8) ∂tR2 +

∫d3x

δ

δφ(x)

(R

2 δS

δφ(x)

)= 0

and this exhibits the unitarity of the theory because it provides that the norm∫[dφ(x)]2Ψ∗Ψ =

∫[dφ(x)]R2 does not depend on time. The quantity R2([φ(x)], t)

represents the probability density for fields to have the configuration φ(x) at time t.

4. WEYL, AND BOHM 73

One can take (4.7) as the starting point for quantization of fields (note exp(iS/)should be single valued). Equations (4.7) and (4.8) suggest a Bohmian interpre-tation with deterministic time evolution given via ∂tφ. Remarkably the statisti-cal predictions of this deterministic interpretation are equivalent to those of theconventional interpretation. All quantum uncertainties are a consequence of theignorance of the actual initial field configuration φ(x, t0). The main reason for theconsistency of this interpretation is the fact that (4.8) with ∂tφ as above represents

of field configurations φ(x) is given by the quantum distribution ρ = R2 at anytime t, provided that ρ is given by R2 at some initial time. The initial distribu-tion is arbitrary in principle but a quantum H theorem explains why the quantumdistribution is the most probable (cf. [954]). Comparing (4.7) with (4.5) we seethat the quantum field satisfies an equation similar to the classical one, with theaddition of a term resulting from the nonlocal quantum potential Q. The quantumequation of motion then turns out to be

(4.9) ∂µ∂µφ +∂V (φ)

∂φ+

δQ

δφ(x; t)= 0

where Q =∫

d3xQ. A priori perhaps the main unattractive feature of the Bohmianformulation appears to be the lack of covariance, i.e. a preferred Lorentz frame isneeded and this can be remedied with the DDW presentation to follow.

Thus one wants a quantum substitute for the classical covariant DDW HJequation (1/2)∂φSµ∂φSµ + V + ∂µSµ = 0. Define then the derivative

(4.10)dA([φ], x)

dφ(x)=∫

d4x′ δA([φ], x′)δφ(x)

where δ/δφ(x) is the spacetime functional derivative (not the space functionalderivative used before in (4.5)). In particular if A([φ], x) is a local functional, i.e.if A([φ], x) = A(φ(x), x) then

(4.11)dA(φ(x), x)

dφ(x)=∫

d4x′ δA(φ(x′), x′)δφ(x)

=∂A(φ(x), x)

∂φ(x)

Thus d/dφ is a generalization of ∂/∂φ such that its action on nonlocal functionalsis also well defined. An example of interest is a functional nonlocal in space butlocal in time so that

(4.12)δA([φ], x′)

δφ(x)=

δA([φ], x′)δφ(x), x0)

δ((x′)0 − x0) ⇒

⇒ dA([φ], x)dφ(x)

=δ

δφ(x, x0)

∫d3x′A([φ],x′, x0)

Now the first equation in (4.3) and the equations of motion become

(4.13)12

dSµ

dφ

dSµ

dφ+ V + ∂µSµ = 0; ∂µφ =

dSµ

dφ

which is appropriate for the quantum modification. Next one proposes a method ofquantization that combines the classical covariant canonical DDW formalism withthe standard specetime asymmetric canonical quantization of fields. The starting

the continuity equation which provides that the statistical distribution ρ([φ(x)], t)

D DONDER,e


point is the relation between the noncovariant classical HJ equation (4.5) and itsquantum analogue (4.7). Suppressing the time dependence of the field in (4.5) wesee that they differ only in the existence of the Q term in the quantum case. Thissuggests the following quantum analogue of the classical covariant equation (4.13)

(4.14)12

dSµ

dφ

dSµ

dφ+ V + Q + ∂µSµ = 0

Here Sµ = Sµ([φ], x) is a functional of φ(x) so Sµ at x may depend on the fieldφ(x′) at all points x′. One can also allow for time nonlocalities (cf. [711]). Thus(4.15) is manifestly covariant provided that Q given by (4.7) can be written in acovariant form. The quantum equation (4.14) must be consistent with the conven-tional quantum equation (4.7); indeed by using a similar procedure to that usedin showing that (4.3) implies (4.5) one can show that (4.14) implies (4.7) providedthat some additional conditions are fulfilled. First S0 must be local in time so that(4.12) can be used. Second Si must be completely local so that dSi/dφ = ∂Si/∂φ,which implies

(4.15) diSi = ∂iS

i + (∂iφ)dSi

dφ

However just as in the classical case in this procedure it is necessary to use thespace part of the equations of motion (4.3). Therefore these classical equations ofmotion must be valid even in the quantum case. Since we want a covariant theoryin which space and time play equal roles the validity of the space part of the (4.3)implies that its time part should also be valid. Consequently in the covariantquantum theory based on the DDW formalism one must require the validity ofthe second equation in (4.13). This requirement is nothing but a covariant versionof the Bohmian equation of motion written for an arbitrarily nonlocal Sµ (thisclarifies and generalizes results in [533]). The next step is to find a covariantsubstitute for the second equation in (4.7). One introduces a vector Rµ([φ], x)which will generate a preferred foliation of spacetime such that the vector Rµ isnormal to the leaves of the foliation. Then define

(4.16) R([φ],Σ) =∫

Σ

dΣµRµ; S([φ], x) =∫

Σ

dΣµSµ

where Σ is a leaf (a 3-dimensional hypersurface) generated by Rµ. Hence thecovariant version of Ψ = Rexp(iS) is Ψ([φ],Σ) = R([φ],Σ)exp(iS([φ],Σ)/). ForRµ one postulates the equation

(4.17)dRµ

dφ

dSµ

dφ+ J + ∂µRµ = 0

In this way a preferred foliation emerges dynamically as a foliation generated bythe solution Rµ of the equations (4.17) and (4.14). Note that Rµ does not play anyrole in classical physics so the existence of a preferred foliation is a purely quantumeffect. Now the relation between (4.17) and (4.7) is obtained by assuming thatnature has chosen a solution of the form Rµ = (R0, 0, 0, 0) where R0 is local intime. Then integrating (4.17) over d3x and assuming again that S0 is local in time

5. QFT AND STOCHASTIC JUMPS 75

one obtains (4.7). Thus (4.17) is a covariant substitute for the second equation in(4.7). It remains to write covariant versions for Q and J and these are

(4.18) Q = − 2

2R

δ2R

δΣφ2(x); J =

R

2δ2S

δΣφ2(x)

where δ/δΣφ(x) is a version of the space functional derivative in which Σ is gen-erated by Rµ. Thus (4.17) and (4.14) with (4.18) represent a covariant substitutefor the functional SE equivalent to (4.8). The covariant Bohmain equations (4.13)imply a covariant version of (4.9), namely

(4.19) ∂µ∂µφ +∂V

∂φ+

dQ

dφ= 0

Since the last term can also be written as δ(∫

d4xQ)/δφ(x) the equation of motion(4.19) can be obtained by varying the quantum action

(4.20) AQ =∫

d4xLQ =∫

d4x(L−Q)

Thus in summary the covariant canonical quantization of fields is given by equa-tions (4.13), (4.14), (4.17), and (4.18). The conventional functional SE correspondsto a special class of solutions for which Ri = 0, Si are local, while R0 and S0 arelocal in time. In [708] a multifield generalization is also spelled out, a toy modelis considered, and applications to quantum gravity are treated. The main resultis that a manifestly covariant method of field quantization based on the DDWformalism is developed which treats space and time on an equal footing. Un-like the conventional canonical quantization it is not formulated in terms of asingle complex SE but in terms of two coupled real equations. The need for aBohmian formulation emerges from the requirement that the covariant methodshould be consistent with the conventional noncovariant method. This suggeststhat Bohmian mechanics (BM) might be a part of the formalism without whichthe covariant quantum theory cannot be formulated consistently.

5. QFT AND STOCHASTIC JUMPS

The most extensive treatment of Bohmian theory is due to a group based inGermany, Italy, and the USA consisting of V. Allori, A. Barut, K. Berndl, M.Daumer, D. Durr, H. Georgi, S. Goldstein, J. Lebowitz, S. Teufel, R. Tumulka,and N. Zanghi (cf. [1, 26, 88, 102, 103, 104, 105, 288, 324, 325, 326, 327,328, 329, 330, 414, 415, 437, 417, 418, 415, 416, 417, 418, 419, 927, 928,948]). There is also of course the pioneering work of deBroglie and Bohm (seee.g. [154, 126, 127, 128, 129, 154]) as well as important work of many otherpeople (cf. [68, 94, 95, 110, 138, 148, 164, 165, 166, 186, 187, 188, 189,191, 197, 198, 236, 277, 295, 298, 305, 306, 346, 347, 373, 374, 375, 438,472, 471, 474, 478, 479, 480, 873, 905, 953, 961]). We make no attemptto survey the philosophy of Bohmian mechanics (BM), or better deBroglie-Bohmtheory (dBB theory), here. This involves many issues, some of them delicate,which are discussed at length in the references cited. The book [111] by Hollandprovides a good beginning and in view of recent work perhaps another book onthis subject alone would be welcome. There is a lot of associated “philosophy”,


involving hidden variables, nonlocality, EPR ideas, wave function collapse, pilotwaves, implicate order, measurement problems, decoherence, etc., much of whichhas been resolved or might well be forgotten. Many matters are indeed clarifiedalready in the literature above (cf. in particular [94, 95, 330, 415, 471]) and wewill not belabor philosophical matters. It may well be that a completely unifiedmathematical theory is beyond reach at the moment but thre are already quiteaccurate and workable models available and the philosophy of dBB theory asdeveloped by the American-German-Italian school mentioned is quite sophisticatedand convincing.

Basically, following [415], for the nonrelativistic theory, GM for N particles isdetermined by the two equations

(5.1) iψt = Hψ;dqk

dt=

2mk[ψ∗∂kψ

ψ∗ψ

]The latter equation is called the guidance or pilot equation which choreographsthe motion of the particles. If ψ is spinor valued the products in the numeratorand denominator are scalar products and if external magnetic fields are presentthe gradient ∇ ∼ (∂k) should be understood as the covariant derivative involvingthe vector potential (thus accomodating some versions of field theory - more onthis later). Since the denominator vanishes at nodes of ψ existence and uniquenessof solutions for Bohmian dynamics is nontrivial but this is proved in [104, 106].This formula extends to spin and the right side corresponds to J/ρ which is theratio of the quantum probability current to the quantum probability density. Fur-ther from the quantum continuity condition ∂tρ + div(J) = 0 (derivable from theSE) it follows that if the configuration of particles is random at the initial time t0with probability distribution ψ∗ψ then this remains true for all times (assumingno interaction with the environment). Upon setting ψ = Rexp(iS/) one identi-fies pk = mkvk with ∂kS (which is equivalent to the guiding equation for particleswithout spin) and this corresponds to particles being acted upon by the force ∂kQgenerated by the quantum potential (in addition to any “classical” forces).

REMARK 2.5.1. Recall from Section 2.3.2 that in the BFM theory ofBohmian type q = p/mQ = p/m where mQ = m(1−∂EQ) in stationary situationswith energy E. Here one is using a Floydian time and there has been a great dealof discussion, involving e.g. tunneling times (see e.g. [138, 139, 140, 191, 194,197, 198, 305, 306, 307, 296, 309, 347, 373, 374, 375, 376, 520]). We donot attempt to resolve any issues here and refer to the references for up to dateinformation.

In any event we proceed with BM or dBB theory in full confidence not onlythat it works but that it is probably the best way to look at QM. We regardthe quantum potential Q as being a quantization vehicle which expresses the in-fluence of quantum fluctuations (cf. Chapters 1,4,5); it also arises in describingWeyl curvature (cf. Chapters 4,5) and thus we regard it as perhaps the funda-mental object of QM. Returning now to [415] one notes that the predictions of


BM for measurements must agree with those of standard QM provided configura-tions are random with distributions given by the quantum equilibrium distribution|ψ|2. Then a probability distribution ρψ depending on ψ is called equivariant if(ρψ)t = ρψt where the right side comes from the SE and the left from the guidingequation (since ρt + div(J) = 0 with v = J/ρ arises in (5.1)). This has beenstudied in detail and we summarize some results below. Further BM can han-dle spin via (5.1) (as mentioned above) and nonlocality is no problem; howeverLorentz invariance, even for standard QM, is tricky and one views it as an emer-gent symmetry. Further QFT with particle creation and annihilation is a currenttopic of research (cf. Sections 2.3 and 2.4) and some additional remarks in thisdirection will follow. The papers [324, 325] are mainly about quantum equi-librium, absolute uncertainty, and the nature of operators. There are two longpapers here (75 and 77 pages) and an earlier paper of 35 pages so we make noattempt to cover this here. We mention briefly some results of the two more re-cent papers however. Thus from the abstract to the second paper of [324] thequantum formalism is treated as a measurement formalism, i.e. a phenomenolog-ical formalism describing certain macroscopic regularities. One argues that it canbe regarded and best be understood as arising from Bohmian mechanics, whichis what emerges from the SE for a system of particles when one merely insiststhat “particles’ means particles. BM is a fully deterministic theory of particles inmotion, a motion choreographed by the wave function. One finds that a Bohmianuniverse, although deterministic, evolves in such a manner that an appearance ofrandomness emerges, precisely as described by the quantum formalism and givenby ρ = |ψ|2. A crucial ingredient in the analysis of the origin of this randomnessis the notion of the effective wave function of a subsystem. When the quantumformalism is regarded as arising in this way the paradoxes and perplexities so oftenassociated with (nonrelativistic) quantum theory evaporate. A fundamental facthere is that given a SE iψt = −(2/2)

∑(∆kψ/mk)+V ψ one can derive a velocity

formula vψk = (/mk)(∇kψ/ψ) by general arguments based on symmetry consid-

erations and this yields (5.1) without any recourse to a formula ψ = Rexp(iS/).Further the continuity equation ρt + div(ρvψ) = 0 holds and this implies theequivariance ρ(q, t) = |ψ(q, t)|2 provided this is true at (t0, q0). The distributionρ = |ψ|2 is called the quantum equilibrium distribution (QELD) and a system isin quantum equilibrium when the QELD is appropriate for its description. Thequantum equilibrium hypothesis (QEH) is that if a system has wave function ψthen ρ = |ψ|2. It is necessary to discuss wave functions of systems and subsystemsat some length and it is argued that in a universe governed by BM it is impossibleto know more about the configuration of any subsystem than what is expressedvia ρ = |ψ|2 (despite the fact that for BM the actual configuration is an objectiveproperty, beyond the wave function). Moreover, this uncertainty, of an absoluteand precise character, emerges with complete ease, the structure of BM being suchthat it allows for the formulation and clean demonstration of statistical statementsof a purely objective character which nontheless imply the claims concerning theirreducible limitations on possible knowledge, whatever this knowledge may pre-cisely mean and however one might attempt to obtain this knowledge, provided itis consistent with BM. This limitation on what can be known is called absolute


uncertainty. One proceeds by analysis of systems and subsystems and we refer to[324] for details. In [325] one shows how the entire quantum formalism, operatorsas observables, etc. naturally emerges in BM from the analysis of measurments.It is however quite technical, with considerable important and delicate reasoning,and we cannot possibly deal with it in a reasonable number of pages.

We go to [326] now where a comprehensive theory is developed for Bohmianmechanics and QFT (cf. also [330]). Bohm and subsequently Bell had proposedsuch models and the latter is model is modified and expanded in [326, 330] inthe context of what are called Bell models. One will treat the configuration spacevariables in terms of Markov processes with jumps (which is reminiscent of thediffusion picture in [673, 674] (cf. also [186]). Roughly one thinks of world linesinvolving particle creation and annihilation, hence jumps, and writes Q = ∪∞

0 Qn

where, taking identical particles, the sector Qn is best defined as R3n/Sn whereS ∼ permutations. For several particle species one forms several copies of Q, onefor each species, and obtains a union of sectors Q(n) where now n ∼ (n1, . . . , nk) forthe k species of particles. Note that a path Q(t) will typically have discontinuities,even if there is nothing discontinuous in the world line pattern, because it jumpsto a different sector at every creation or annihilation event. One can think of thebosonic Fock space as a space of L2 functions on ∪nR3n/Sn with the fermionicFock space being L2 functions on ∪nR3n, antisymmetric under permutation. ABell type QFT specifies such world line patterns or histories in configuration spaceby specifying three sorts of “laws of motion”: when to jump, where to jump, andhow to move between jumps. One consequence of these laws (to be enumerated)is the property of preservation of |Ψ0|2 at time t0 to be equal to |Ψt|2 at timet; this is called equivariance (see above and cf. [325, 327] for more detail onequivariance for Bohmian mechanics - the same sort of reasoning will apply here).One will use the quantum state vector Ψ to determine the laws of motion andhere a state described by the pair (Ψt, Qt) where Ψ evolves according to the SEi∂tΨt = HΨ. Typically H = H0 + HI and it is important to note that althoughthere is an aactual particle number N(t) = #Q(t) or Q(t) ∈ QN(t), Ψ need not bea number eigenstate (i.e. concentrated in one sector). This is similar to the usualdouble-slit experiment in which the particle passes through only one slit althoughthe wavefunction passes through both. As with this experiment, the part of thewave function that passes through another sector of Q (or another slit) may wellinfluence the behavior of Q(t) at a later time. The laws of motion of Qt dependon Ψt (and on H) and the continuous part of the motion is governed by

(5.2)dQt

dt= vΨt(Qt) = Ψ∗

t (Qt)( ˙qΨt)(Qt)Ψ∗

t (Qt)Ψt(Qt); ˙q =

d

dτeiH0τ/

∣∣∣∣τ=0

=i

[H0, q]

Here ˙q is the time derivative of the Q valued Heisenberg position operator q evolvedwith H0 alone. One should understand this as saying that for any smooth functionf : Q→ R

(5.3)df(Qt)

dt= Ψ∗

t (Qt)(i/)[H0, f ]Ψt)(Qt)Ψ∗

t (Qt)Ψt(Qt)

This expression is of the form vΨ ·∇f(Qt) (as it must be for defining a dynamics forQt) if the free Hamiltonian is a differential operator of up to second order (more onthis later). Note that the KG equation is not covered by (5.2) or (5.3). Thenumerator and denominators above involve, when appropriate, scalar products inspin space. One may view v as a vector field on Q and thus as consisting of onevector field vn on every manifold Qn; it is then vN(t) that governs the motionof Q(t) in (5.2). If H0 were the Schrodinger operator −

∑n1 (2/2m)∆i + V (5.2)

yields the Bohm velocities vΨi = (/mi)[Ψ∗∇iΨ/Ψ∗Ψ]. When H0 is the “second

quantization” of a 1-particle Schrodinger operator (5.2) involves equal masses inevery sector Qn. Similarly in case H0 is the second quantization of the Diracopertor −icα · ∇+ βmc2 (5.2) says that a configuration Q(t) (with N particles)moves according to (the N-particle version of) the known variant of Bohm’s velocityformula for Dirac wavefunctions vΨ = (Ψ∗αΨ/Ψ∗Ψ)c (cf. [127]). The jumps noware stochastic in nature, i.e. they occur at random times and lead to randomdestinations. In Bell type QFT God does play dice. There are no hidden variableswhich would fully determine the time and destination of a jump (cf. here Section2.3 and the effectivity parameters). The probability of jumping, with the next dtseconds to the volume dq in Q is σΨ(dq|Qt)dt with

(5.4) σΨ(dq|q′) =2

[Ψ∗(q) < q|HI |q′ > Ψ(q′)]+

Ψ∗(q′)Ψ(q′)dq

where x+ = max(x, 0). Thus the jump rate σΨ depends on the present configu-ration Qt, on the state vector Ψt which has a guiding role similar to that in theBohm theory, and of course on the overall setup of the QFT as encoded in theinteraction Hamiltonian HI (cf. [326] for a simple example). There is a strikingsimilarity between (5.4) and (5.2) in that they are both cases of “minimal” Markovprocesses associated with a given Hamiltonian (more on this below). When H0 isreplaced by HI in the right side of (5.3) one obtains an operator on functions f(q)that is naturally associated with the process defined by the jump rates (5.4).

The field operators (operator valued fields on spacetime) provide a connection,the only connection in fact, between spacetime and the abstract Hilbert space con-taining the quantum states |Ψ >, which are usually regarded not as functions butas abstract vectors. What is crucial now is that (i) The field operators naturallycorrespond to the spatial structure provided by a projection valued (PV) measureon configuration space Q, and (ii) The process defined here can be efficiently ex-pressed in terms of a PV measure. Thus consider a PV measure P on Q acting onH where for B ⊂ Q, P (B) means the projection to the space of states localizedin B. Then one can rewrite the formulas above in terms of P and |Ψ > and we get

(5.5)df(Qt)

dt=

< Ψ|P (dq) i [H0, f ]|Ψ >

< Ψ|P (dq)|Ψ >

∣∣∣∣∣q=Qt

; f =∫

q∈Q

f(q)P (dq)

(for smooth functions f : Q→ R) and

(5.6) σΨ(dq|q′) =2

< Ψ|P (dq)HIP (dq′)|Ψ >]+

< Ψ|P (dq′)|Ψ >

Note that < Ψ|P (dq)|Ψ > is the probability distribution analogous to the standard|Ψ(q)|2dq. The next question is how to obtain the PV measure P from the fieldoperators. Such a measure is equivalent to a system of number operators (moreon this below); thus an additive operator valued set function N(R), R ∈ R3 suchthat the N(R) commute pairwise and have spectra in the nonnegative integers.By virtue of the canonical commutation and anticommutation relations for thefield operators φ(x) the easiest way to obtain such a system of number operatorsis via N(R) =

∫R

φ∗(x)φ(x)d3x. Thus what one needs from a QFT in order toconstruct trajectories are: (i) a Hilbert space H (ii) a Hamiltonian H = H0 + HI

(iii) a configuration space Q (or measurable space), and (iv) a PV measure on Q

acting on H. This will be done below following [326].

We go now to the last paper in [326] which is titled quantum Hamiltonians andstochastic jumps. The idea is that for the Hamiltonian of a QFT there is associateda |Ψ|2 distributed Markov process, typically a jump process, on the configurationspace of a variable number of particles. A theory is developed generalizing workof J. Bell and the authors of [326]. The central formula of the paper is

(5.7) σ(dq|q′) =[(2/) < Ψ|P (dq)HP (dq′)|Ψ >]+

< Ψ|P (dq′)|Ψ >

It plays a role similar to that of Bohm’s equation of motion

(5.8)dQ

dt= v(Q); v = Ψ∗∇Ψ

Ψ∗ΨTogether these two equations make possible a formulation of QFT that makesno reference to observers or measurements, while implying that observers, whenmaking measurements, will arrive at precisely the results that QFT is knownto predict. This formulation takes up ideas from the seminal papers of J. Bell[94, 95] and such theories will be referred to as Bell-type QFT’s. The aim is topresent methods for constructing a canonical Bell type model for more of less anyregularized QFT. One assumes a well defined Hamiltonian as given (with cutoffsincluded if needed). The primary variables of such theories are particle positionsand Bell suggested a dynamical law governing the motion of the particles in whichthe Hamiltonian H and the state vector Ψ determine the jump rates σ. Theserates are in a sense the smallest choice possible (explained below) and are calledminimal jump rates; they preserve the |Ψ|2 distribution. Bell type QFT’s can alsobe regarded as extensions of Bohmian mechanics which cover particle creationand annihilation; the quantum equilibrium distribution more or less dictates thatcreation of a particle occurs in a stochastic manner as in the Bell model. We recallthat for Bohmian mechanics in addition to (5.8) one has an evolution equationi∂tΨ = HΨ for the wave function with H = −(2/2∆ + V for spinless particles(∆ = div∇). For particles with spin Ψ takes values in the appropriate spin spaceCk, V may be matrix valued, and inner products in (1.66) are understood asinvolving inner products in spin spaces. The success of the Bohmian method isbased on the preservation of |Ψ|2, called equivariance and this follows immediatelyfrom comparing the continuity equation for a probability distribution ρ associatedwith (5.8), namely ∂tρ = −div(ρv), with the equation for |Ψ|2 following from the

SE, namely

(5.9) ∂t|Ψ|2(q, t) = (2/)[Ψ∗(q, t)(HΨ)(q, t)]

In fact it follows from the continuity equation that

(5.10) (2/)[Ψ∗(q, t)(HΨ)(q, t)] = −div[Ψ∗(q, t)∇Ψ(q, t)]

so recalling (5.8), one has ∂t|Ψ|2 = −div(|Ψ|2v), and hence if ρ+|Ψt|2 as some timet there results ρ = |ψt|2 for all times. One is led naturaly to the consideration ofMarkov processes as candidates for the equivariant motion of the configuration Qfor more general Hamiltonians (see e.g. [506, 674, 810, 815] for Markov processes- [674] is especially good for Markov processes with jumps and dynamics but wefollow [506, 815] for background since the ideas are more or less clearly statedwithout a deathly deluge of definitions and notation - of course for a good theorymuch of the verbiage is actually important).

DEFINITION 5.1. Let (E be a Borel σ-algebra of subsets of E. For Ωgenerally a path space (e.g. Ω ∼ C(R+, E) with Xt(ω) = X(t, ω) = ω(t) andFt = σ(Xs(ω), s ≤ t) a filtration by Borel sub σ-algebras) a Markov process(C11) X = (Ω, Ft, Xt, Pt, P x, xE) with t ≥ 0 and state space (E,E), isan E valued stochastic process adapted to Ft such that for 0 ≤ s ≤ t, f ∈ bE(bE means bounded E measurable functions), and x ∈ E, Ex[f(Xs+t)|Ft] =(Ptf)(Xs), P xae (ae means almost everywhere). Here Pt is a transition functionon (E,E), i.e. a family of kernels Pt : E × E→ [0, 1] such that

(1) For t ≥ 0 and x ∈ E, Pt(x, ·) is a measure on E with Pt(x,E) ≤ 1(2) For t ≥ 0 and Γ ∈ E Pt(·,Γ) is E measurable(3) For x, t ≥ 0, x ∈ E, and Γ ∈ E one has Pt+s(x,Γ) =

∫E

Ps(x, dy)Pt(y, Γ)

The equation in #3 is called the Chapman-Kolmogorov (CK) equation and, think-ing of the transition functions as inducing a family Pt of positive bounded op-erators or norm less than or equal to 1 on bE one has Ptf(x) = (Ptf)(x) =∫

EPt(x, dy)f(y) in which case the CK equation has the semigroup property PsPt =

Ps+t for s, t ≥ 0.

Under mild regularity conditions if a transition semigroup Pt is given therewill exist on some probability space a Markov process X with suitable paths suchthat the strong Markov property holds, i.e. Ex[f(XS+t)]|Fx] = (Ptf)(XS) P x aewhenever S is a finite stopping time (here S : Ω → [0,∞] is a E stopping time ifS ≤ t = ω; S(ω) ≤ t ∈ Et for every t < ∞).

EXAMPLE 5.1. A Markov process with countable state space is called aMarkov chain. One writes pij(t) = Pt(i, j) with P (t) = pij(t); i, j ∈ E.Assume Pt(i, E) = 1 and pij(t) → δij as t ↓ 0. This will imply that in factp′ij(0) = qij exists and the matrix Q = (qij) is called an infinitesimal generatorof Pt with qij ≥ 0 (i = j) and

∑j qik = 0 (i ∈ E). This illustrates some

important structure for Markov processes. Thus when P ′(0) = Q exists one canwrite P ′(t) = limε→0ε

−1[P (t + ε)− P (t)] = limε→0P (t)[P (ε)− I] = P (t)Q. Thensolving this equation one has P (t) = exp(tQ as a semigroup generated by Q. The

resolvent is defined via Rλ =∫∞0

exp(−λt)Ptdt and one can regard it as

(5.11) (λRλ)ij =∫ ∞

0

λexp(−λt)pij(t)dt = P(XT = j|X0 = i)

where T is a random variable independent of X with the exponential distributionof rate λ. It follows then that Rλ = (λ−Q)−1 and Rλ−Rµ = (µ− λ)RλRµ. Thewhole subject is full of pathological situations however and we make no attemptto describe this.

REMARK 2.5.2. [674] is oriented toward diffusion processes and departsfrom the concept that the kinematics of quantum particles is stochastic calculus (inparticular Markov processes) while the kinematics of classical particles is classicaldifferential calculus. The relation between these two calculi must be established.Thus classically x(t) = x(a)+

∫ t

av(s, x(s))ds while for a particle with say Brownian

noise Bt and a drift field a(t, x(t)) one has

(5.12) Xt = Xa +∫ t

a

a(s, Xs)ds +∫ t

a

σ(s, Xs)dBs

We recall dB2t ∼ dt so Xt has no velocity and the drift field a(t, x) is not an average

speed. However P [X] =∫Ω

XdP is the expectation (since P [σ(t,Xt)dBt] = 0).Now the notation for Markov processes involves nonnegative transition functionsP (s, x; t, B) with a ≤ s ≤ t ≤ b, x ∈ Rd, and B ∈ B(Rd) which are measures inB, measurable in x, and satisfy the CK equation

(5.13) P (s, x; t, B) =∫Rd

P (s, x; r, dy)P (r, y; t, B); P (s, x; t,Rd) = 1

If there is a measurable function p such that P (s, x; t, B) =∫

Bp(s, x; t, y)dy (t −

s > 0) then p is a transition density. One defines a probability measure P on apath space Ω = (Rd)[a,b] via finite dimensional distributions(5.14)

P [f(Xa, Xt1 , · · · , Xtn, Xb)] =

∫µa(dx0)P (a, x0; t1, dx1)P (t1, x1; t2, dx2) · · · ×

× · · ·P (tn−1, xn−1; b, dxn)f(x0, · · · , xn)Moreover one defines a family Xt; t ∈ [a, b] on Ω via Xt(ω) = ω(t), ω ∈ Ω. Noteone assumes the right continuity of Xt(ω) ae. This representation can be writtenas P = [µaP >> and is called the Kolmogorov representation of P. Let now Ft

sbe a filtration as before, i.e. a family of σ-fields generated by Xr(ω); s ≤ r ≤ t.Then we have a Markov process Xt, t ∈ [a, b],Ft

s, P. Replacing µa by δx anda by s with s < t1 < · · · < tn−1 < tn ≤ b one defines probability measuresP(s,x), (s, x) ∈ [a, b]×Rd from (1.67) via

(5.15) P(s,x)[f(Xt1 , · · · , Xtn−1 , Xtn)] =

=∫

P (s, x; t1, dx1) · · ·P (tn−1, xn−1; tn, dxn)f(x1, · · · , xn)

As a special case one has P(s,x)[f(t,Xt)] =∫

P (s, x; t, dy)f(t, y) and one can alsoprove that P [GF ] = P [GP(s,Xs)[F ]] for any bounded Fs

a measurable G and anybounded Fb

s measurable F. This is the time inhomogeneous Markov property which


can be written in terms of conditional expectations as P [F |Fsa] = P(s,Xs)[F ], P ae.

There is a great deal of material in [674] about Markov processes with jumps butwe prefer to stay here with [326] for notational convenience.

Going back to [326] we consider a Markov process Qt on configuration spacewith transition probabilities characterized by the backward generator Lt, a timedependent linear operator acting on functions f via Ltf(q) = (d/ds)E(f(Qt+s|Qt =q) where d/ds means the right derivative at s = 0 and E(·|·) is conditional expec-tation. Equivalently the transition probabilities are characterized by the forwardgenerator Lt (or simply generator) which is also a linear operator but acts on(signed) measures on the configuration space. Its defining property is that forevery process Qt with the given transition properties ∂tρt = Ltρt. Thus L is dualto Lt in the sense

(5.16)∫

f(q)Ltρ(dq) =∫

Ltf(q)ρ(dq)

Given equivariance for |Ψ|2, one says that the corresponding transition probabil-ities are equivariant and this is equivalent to Lt|Ψ|2 = ∂y|Ψ|2 for all t; when thisholds one says that Lt is an equivariant generator (with respect to Ψt and H).One says that a Markov process is Q equivariant if and only if for every t thedistribution ρt of Qt equals |Ψt|2. For this equivariant transition probabilities arenecessary but not sufficient; however for equivariant transition probabilities thereis a unique equivariant Markov process. The crucial idea here for construction ofan equivariant Markov process is to note that (5.9) is completely general and tofind a generator Lt such that the right side of (5.9) can be read as the action of Lon ρ = |Ψ|2 means (2/)Ψ∗HΨ = L|Ψ|2. This will be implemented later. For Hof the form −(2/2)∆ + V one has (5.10) and hence

(5.17)2Ψ∗HΨ = −div(Ψ∗∇Ψ) = −div

(|Ψ|2Ψ∗∇Ψ

|Ψ|2

)Since the generator of the (deterministic) Markov process corresponding to the dy-namical system dQ/dt = v(Q) is given by a velocity vector field is Lρ = −div(ρv)we may recognize the last term of (1.67) as L|Ψ|2 with L the generator of thedeterministic process defined by (5.8). Thus Bohmian mechanics arises as thenatural equivariant process on configuration space associated with H and Ψ. Onenotes that Bohmian mechanics is not the only solution of (2/)Ψ∗HΨ − L|Ψ|2;there are alternatives such as Nelson’s stochastic mechanics (and hence Nagasawa’stheory of [672, 674]) and other velocity formulas (cf. [295]).

For equivariant jump processes one says that a (pure) jump process is a Markovprocess on Q for which the only motion that occurs is via jumps. Given that Qt = qthe probability for a jump to q′ (i.e. into the infinitesimal volume dq′ around q′) bytime t+dt is σt(d′q|q)dt where σ is called the jump rate. Here σ is a finite measurein the first variable; σ(B|q) is the rate (i.e. the probability per unit time) of jump-ing to somewhere in the set B ⊂ Q given that the present location is q. The overalljump rate is σ(Q|q) (sometimes one writes ρ(dq) = ρ(q)dq). A jump first occurswhen a random waiting time T has elapsed, after the time t0 at which the process

was started or at which the most recent previous jump has occured. For purposesof simulating or constructing the process, the destination q′ can be chosen at thetime of jumping, t0 + T , with probability distribution σt0+T (Q|q)−1σt0+T (·|q). Incase the overall jump rate is time independent T is exponentially distributed withmean of σ(Q|q)−1. When the rates are time dependent (as they will typically inwhat follows) the waiting time remains such that

∫ t0+T

t0σt(Q|q)dt is exponentially

distributed with mean 1, i.e. T becomes exponential after a suitable (time depen-dent) rescaling of time. The generator of a pure jump process can be expressed interms of the rates

(5.18) Lρ(dq) =∫

q′∈Q

(σ(dq|q′)ρ(dq′)− σ(dq′)|q)ρ(dq))

which is a balance or master equation expressing ∂tρ as the gain due to jumpsto dq minus the loss due to jumps away from dq. One says the jump rates areequivariant if Lσ is an equivariant generator.

Given a Hamiltonian H = H0 + HI one obtains

(5.19) (2/)Ψ∗H0Ψ + (2/)Ψ∗HIΨ− L|Ψ|2

This opens the possibility of finding a generator L = L0+LI given (2/)Ψ∗H0Ψ =L0|Ψ|2 and (2/)Ψ∗HIΨ = LI |Ψ|2; this will be called process additivity andcorrespondingly L = L0 + LI . If one has two deterministic processes of the formLρ = −div(ρv) then adding generators corresponds to v = v+v2. For a pure jumpprocess adding generators corresponds to adding rates σi which is equivalent tosaying there are two kinds of jumps. Now add generators for a deterministic anda jump process via

(5.20) Lρ(q) = −div(ρv)(q) +∫

q′∈Q

(σ(q|q′)ρ(q′)− σ(q′|q)ρ(q)) dq′

This process moves with velocity v(q) until it jumps to q′ where it continuesmoving with velocity v(q′). One can understand (5.20) in terms of gain or loss ofprobability density due to motion and jumps; the process is piecewise deterministicwith random intervals between jumps and random destinations. Note that for aWiener process the generator is the Laplacian and adding to it the generator of adeterministic process means introducing a drift.

Now consider HI and note that in QFT’s with cutoffs it is usually the casethat HI is an integral operator. Hence one writes here H ∼ HI and thinks of itas an integral operator with Q ∼ Rn. What characterizes jump processes is thatsome amount of probability that vanishes at q ∈ Q can reappear in an entirelydifferent region say at q′ ∈ Q. This suggests that the Hamiltonians for which theexpression (5.9) for ∂t|Ψ|2 is naturally an integral over q′ correspond to pure jumpprocesses. Thus when is the left side of (2/)|psi∗HΨ = L|Ψ|2 an integral overq′ or (HΨ)(q) =

∫dq′ < q|H|q′ > Ψ(q′). In this case one should choose the jump

rates so that when ρ = |Ψ|2 one has

(5.21) σ(q|q′)ρ(q′)− σ(q′|q)ρ(q) = (2/)Ψ∗(q) < q|H|q′ > Ψ(q′)

This suggests, since jump rates are nonnegative and the right side of (5.21) isantisymmetric) that σ(q|q′)ρ(q′) = [(2/)Ψ∗(q) < q|H|q′ > Ψ(q′)]+ or

(5.22) σ(q|q′) =(2/)Ψ∗(q) < q|H|q′ > Ψ(q′)]+

Ψ∗(q′)Ψ(q′)

These rates are an instance of what can be called minimal jump rates associatedwith H (and Ψ). They are actually the minimal possible values given (5.21) andthis is discussed further in [326]. Minimality entails that at any time t one of thetransitions q1 → q2 or q2 → q1 is forbidden and this will be called a minimal jumpprocess. One summarizes motions via

H motionintegral operator jumps

differential operator deterministic continuous motionmultiplication operator no motion (L = 0)

The reasoning above applies to the more general setting of arbitrary configurationspaces Q and generalized observables - POVM’s - defining what the “position”representation is to be. One takes the following ingredients from QFT

(1) A Hilbert space H with scalar product < Ψ|Φ >.(2) A unitary one parameter group Ut in H with Hamiltonian H, i.e. Ut =

exp[−(i/)tH], so that in the Schrodinger picture the state Ψ evolves viai∂tΨ = HΨ. Ut could be part of a representation of the Poincare group.

(3) A positive operator valued measure (POVM) P (dq) on Q acting on H sothat the probability that the system in the state Ψ is localized in dq attime t is Pt(dq) =< Ψt|P (dq)|Ψt >.

Mathematically a POVM on Q is a countably additive set function (measure)defined on measurable subsets of Q with values in the positive (bounded selfadjoint) operators on a Hilbert space H such that P (Q) = Id. Physically forpurposes here P (·) represents the (generalized) position observable, with valuesin Q. The notion of POVM generalizes the more familiar situation of observablesgiven by a set of commuting self adjoint operators, corresponding by means ofthe spectral theorm to a projection valued measure (PVM) - the case where thepositive operators are projection operators (see [326] for discussion). The goalnow is to specify equivariant jump rates σ = σΨ,H,P so that LσP = dP/dt. Tothis end one could take the following steps.

(1) Note that (dPt(dq)/dt) = (2/) < Ψt|P (dq)H|Ψt >.(2) Insert the resolution of the identity I =

∫q′∈Q

P (dq′) and obtain

(5.23) (dPt(dq)/dt) =∫

q′∈Q

Jt(dq, dq′);

Jt(dq, dq′) = (2/) < Ψt|P (dq)HP (dq′)|Ψt >

(3) Observe that Jis antisymmetric so since x = x+ − (−x)+ one has

(5.24) J(dq, dq′) = [(2/) < Ψ|P (dq)HP (dq′)|Ψ]+−

−[(2/) < Ψ|P (dq′)HP (dq)|Ψ >]+

(4) Multiply and divide both terms by P(·) obtaining

(5.25)∫

q′∈Q

J(dq, dq′) =∫

q′∈Q

((2/) < Ψ|P (ddq)HP (dq′)|Ψ >]+

< Ψ|P (dq′)|Ψ >P(dq′)−

− [(2/) < Ψ|P (dq′)HP (dq)|Ψ >]+

< Ψ|P (dq)|Ψ >P(dq)

)(5) By comparison with (5.18) recognize the right side of the above equation

as LσP with Lσ the generator of a Markov jump process with jump rates(5.7) (minimal jump rates).

Note the right side of (5.7) should be understood as a density (Radon-Nikodymderivative).

When H0 is made of differential operators of up to second order one cancharacterize the process associated with H0 in a particularly succinct manner asfollows. Define for any H,P,Ψ an operator L acting on functions f : Q → Rwhich may or may not be the backward generator of a process via

(5.26) Lf(q) = < Ψ|P (dq)Lf |Ψ >

< Ψ|P (dq)Ψ >= < Ψ|P (dq)(i/)[H, f ]|Ψ >

< Ψ|P (dq)|Ψ >

where [ , ] means the commutator and f =∫

q∈Qf(q)P (dq) with L the generator

of the (Heisenberg) evolution f ,

(5.27) Lf = (d/dτ)exp(iHτ/)f exp(−iHτ/)|τ=0 = (i/)[H, f ]

Note if P is a PVM then f = f(q). (5.26) could be guessed in the following manner:Since Lf is in a certain sense the time derivative of f it might be expected to berelated to Lf which is in a certain sense (cf. (5.27)) the time derivative of f . As away of turning the operator Lf into a function Lf(q) the middle term in (5.26) isan obvious possibility. Note also that this way of arriving at (5.26) does not makeuse of equivariance. The formula for the forward generator equivalent to (5.26)reads

(5.28) Lρ(dq) = < Ψ| (dρ/dP)(i/)[H,P (dq)]|Ψ >

Whenever L is indeed a backward generator we call it the minimal free (backward)generator associated with Ψ,H, P . Then the corresponding process is equivariantand this is the case if (and there is reason to expect, only if) P is a PVM and His a differential operator of up to second order in the position representation, inwhich P is diagonal. In that case the process is deterministic and the backwardgenerator has the form L = v · ∇ where v is the velocity field; thus (5.26) directlyspecifies the velocity in the form of a first order differential operator v ·∇. In caseH is the N-particle Schrodinger operator with or without spin (5.26) yields theBohmian velocity (5.8) and if H is the Dirac operator the Bohm-Dirac velocityemerges. Thus in some cases (5.26) leads to just the right backward generator.In [326] there are many examples and mathematical sections designed to provevarious assertions but we omit this here..

6. BOHMIAN MECHANICS IN QFT 87

6. BOHMIAN MECHANICS IN QFT

We extract here from a fascinating paper [713] by H. Nikolic. Quantum fieldtheory (QFT) can be formulated in the Schrodinger picture by using a functionaltime dependent SE but this requires a choice of time coordinate and the corre-sponding choice of a preferred foliation of spacetime producing a relativisticallynoncovariant theory. The problem of noncovariance can be solved by replacingthe usual time dependent SE with the many fingered time (MFT) Tomonaga-Schwinger equation, which does not require a preferred foliation and the quantumstate is a functional of an arbitrary timelike hypersurface. In a manifestly co-variant formulation introduced in [316] the hypersurface does not even have tobe timelike. The paper [713] develops a Bohmian interpretation for the MFTtheory for QFT and refers to [77, 108, 255, 473, 478, 587, 711, 708, 769,772, 774, 819, 820, 876, 910] for background and related information. Thuslet x = xµ = (x0,x) be spacetime coordinates. A timelike Cauchy hypersurfaceΣ can be defined via x0 = T (x) with x denoting coordinates on Σ. Let φ(x) be adynamical field on Σ (a real scalar field for convenience) and write T, φ withoutan argument for the functions themselves with φ = φ|Σ etc. Let H(x) be theHamiltonian density operator and then the dynamics of a field φ is described bythe MFT Tomonaga-Schwinger equation

(6.1) HΨ[φ, T ] = iδΨ[φ, T ]δT (x)

Note δT (x) denotes an infinitesimal change of the hypersurface Σ. The quantityρ[φ, T ] = |Ψ[φ, T ]|2 represents the probability density for the field to have a valueφ on Σ or equivalently the probability density for the field to have a value φ attime T . One can say that φ has a definite value ϕ at some time T0 if

(6.2) Ψ[φ, T0] = δ(φ− ϕ) =∏x∈Σ

δ(φ(x)− ϕ(x))

[713] then provides an important discussion of measurement and contextuality inQM which we largly omit here in order to go directly to the Bohmian formulation.

For simplicity take a free scalar field with

(6.3) H(x) = −12

δ2

δφ2(x)+

12[(∇φ(x))2 + m2φ2(x)]

Writing Ψ = Rexp(iS) with R and S real functionals the complex equation (6.1)is equivalent to two real equations with

(6.4)12

(δS

δφ(x)

)2

+12[(∇φ(x))2 + m2φ2(x)] + Q(x, φ, T ] +

δS

δT (x)= 0;

δρ

δT (x)+

δ

δφ(x)

(ρ

δS

δT (x)

)= 0; Q(x, φ, T ] = − 1

2R

δ2R

δφ2(x)

The conservation equation shows that it is consistent to interpret ρ[φ, T ] as theprobability density for the field to have the value φ at the hypersurface determined


by the time T. Now let σx be a small region around x and define the derivative

(6.5)∂

∂T (x)= lim

σx→0

∫σx

d3xδ

δT (x)

where σx → 0 means that the 3-volume goes to zero (note ∂T (y)/∂T (x) = δxy).It is convenient to integrate (6.4) inside a small σx leading to

(6.6)∂ρ

∂T (x)+

∂

∂φ(x)

(ρ

δS

δφ(x)

)= 0

where ∂/∂φ(x) is defined as in (6.5). The Bohmian interpretation consists nowin introducing a deterministic time dependent hidden variable such that the timeevolution of this variable is consistent with the probabilistic interpretation of ρ.From (6.6) one sees that this is naturally achieved by introducing a MFT fieldΦ(x, T ] that satisfies the MFT Bohmian equations of motion

(6.7)∂Φ(x, T ]∂T (x)

=δS

δφ(x)

∣∣∣∣φ=Φ

From (6.7) and the quantum MFT HJ equation (6.4) results

(6.8)

[(∂

∂T (x)

)2

−∇2x + m2

]Φ(x, T ] = − ∂Q(x, φ, T ]

∂ψ(x)

∣∣∣∣φ=Φ

This can be viewed as a MFT KG equation modified with a nonlocal quantumterm on the right side. The general solution of (6.7) has the form

(6.9) Φgen(x), T ] = F (x, c(x, T ];T ]

where F is a function(al) that depends on the right side of (6.7) and c(x, T ] is anarbitrary function(al) with the property

(6.10)δc(x, T ]δT (x)

= 0

This quantity can be viewed as an arbitrary MFT integration constant - it isconstant in the sense that it does not depend on T (x), but it may depend on T atother points x′ = x. To provide the correct classical limit (indicated below) onerestricts c(x, T ] to satisfy

(6.11) c(x, T ] = c(x)

where c(x) is an arbitrary function. Here it is essential to realize that Φ(x, T ] isa function of x but a functional of T; the field Φ depends not only on (x, T (x)) ≡(x, x0) ≡ x but also on the choice of the whole hypersurface Σ that contains thepoint x. Consequently the MFT Bohmian interpretation does not in general assigna value of the field at the point x unless the whole hypersurface containing x isspecified. On the other hand if e.g. δS/δφ(x) on the right side in (6.7) is a localfunctional, i.e. of the form V (x, φ(x

¯), T (x), then the solution of (6.7) is a local

functional of the form

(6.12) Φ(x, T (x) = Φ(x, x0) = Φ(x)

This occurs for example when the wave functional is a local product Ψ[φ, T ] =∏x ψx(φ(x, T (x). Interactions with the measuring apparatus can also produce

6. BOHMIAN MECHANICS IN QFT 89

locality. As for the classical limit one can formulate the classical HJ equation asa MFT theory (cf. [819, 820]) without of course the Q term. Hence by imposinga restriction similar to (6.11) the solution S[φ, T ] can be chosen so that δS/δφ(x)is a local functional; the restriction (6.11) again implies that the classical solutionΦ is also a local functional.

The MFT formalism was introduced by Tomonaga and Schwinger to providethe manifest covariance of QFT in the interaction picture. The picture here isso far not manifestly covariant since time is not treated on an equal footing withspace. However the MFT formalism can be here also in a manifestly covariantmanner via [316, 819, 820]. One starts by introducing a set of 3 real parameterss1, s2, s3 ≡ s to serve as coordinates on a 3-dimensional manifold (a priori sis not related to x). The 3-dimensional manifold Σ can be embedded in the4-dimensional spacetime by introducing 4 functions Xµ(s) and a 3-dimensionalhypersurface is given via xµ = Xµ(s). The 3 parameters si can be eliminatedleading to an equation of the form f(x0, x1, x2, x3) = 0 and assuming that thebackground spacetime metric gµν(x) is given the induced metric qij(s) on thishypersurface is

(6.13) qij(s) = gµν(X(s))∂Xµ(s)

∂si

∂Xν(s)∂sj

Similarly a normal to the surface is

(6.14) n(s) = εµαβγ∂Xα

∂s1

∂Xβ

∂s2

∂Xγ

∂s3

and the unit normal transforming as a spacetime vector is

(6.15) nµ(s) =gµν nν√|gαβnαnβ |

Now some of the original equations above can be written in a covariant form bymaking the replacements

(6.16) x→ s;δ

δT (x)→ nµ(s)

δ

δXµ(s)

The Tomonaga-Schwinger equation (6.1) becomes

(6.17) H(s)Ψ[φ,X] = inµ(s)δΨ[φ,X]δXµ(s)

For free fields the Hamiltonian density operator in curved spacetime is

(6.18) H =−1

2|q|1/2

δ2

δφ2(s)+|q|1/2

2[−qij(∂iφ)(∂jφ) + m2φ2]

The Bohmian equations of motion (6.7) becomes

(6.19)∂Φ(s, T ]∂τ(s)

=1

|q(s)|1/2

δS

δφ(s)

∣∣∣∣φ=Φ

;∂

∂τ(s)≡ limσx→0

∫σx

d3snµ(s)δ

δXµ(s)


Similarly (6.8) becomes

(6.20)

[(∂

∂τ(s)

)2

+∇i∇i + m2

]Φ(s, X] = − 1

|q(s)|1/2

∂Q(s, φ,X]∂φs)

∣∣∣∣φ=Φ

where ∇i is the covariant derivative with respect to si and

(6.21) Q(s, φ,X] = − 1|q(s)|1/2

12R

δ2R

δφ2(s)corresponding to a quantum potential. The same hypersurface Σ can be para-metrized by different sets of 4 functions Xµ(s) of course but quantities such asΨ[φ,X] and Φ(s, X] depend on Σ, not on the parametrization. The freedom inchoosing functions Xµ(s) is sort of a gauge freedom related to the covariance. Nowto find a solution of the covariant equations above it is convenient to fix a gaugeand for a timelike surface the simplest choice is Xi(s) = si. This implies δX(s) = 0which leads to equations similar to those obtained previously. For example (6.19)becomes

(6.22) (g00(x))1/2 ∂Φ(x, X0]∂X0(x)

=1

|q(x)|1/2

δS

δφ(x)

∣∣∣∣φ=Φ

which is the curved spacetime version of (6.7).

CHAPTER 3

GRAVITY AND THE QUANTUM POTENTIAL

Just as we plunged into QM in Chapters 1 and 2 we plunge again into generalrelativity (GR), Weyl geometry, Dirac-Weyl (DW) theory, and deBroglie-Bohm-Weyl (dBBW) theory. There are many good books available for background ingeneral relativity, especially [69] (marvelous for conceptual purposes and for amodern perspective) and [12] (a classic masterpiece with all the indices in theirplace). In addition we mention some excellent books and papers which will arisein references later, namely [52, 121, 351, 458, 498, 551, 657, 715, 723, 819,910, 972]. To develop all the background differential geometry requires a bookin itself and the presentation adopted here will in fact include all this implicitlysince the topics range over a fairly wide field (see also Chapter 5 where cosmologyplays a more central role).

1. INTRODUCTION

A complete description of necessary geometric ideas appears in [657] forexample and we only make some definitions and express some relations here,usng the venerable tensor notation of indices, etc., since even today much of thephysics literature appears in this form. For differential geometry one can refer to[134, 276, 998]. First we give some background on Weyl geometry and Brans-Dicke theory following [12]; for differential geometry we use the tensor notationof [12] and refer to e.g. [121, 358, 458, 498, 723, 731, 972, 998] for othernotation (see also [990] for an interesting variation). One thinks of a differentialmanifold M = Ui, φi with φ : Ui → R4 and metric g ∼ gijdxidxj satisfy-ing g(∂k, ∂) = gk =< ∂k, ∂ >= gk. This is for the bare essentials; one canalso imagine tangent vectors Xi ∼ ∂i and dual cotangent vectors θi ∼ dxi, etc.Given a coordinate change xi = xi(xj) a vector ξi transforming via ξi =

∑∂ix

jξj

is called contravariant (e.g. dxi =∑

∂j xidxj). On the other hand ∂φ/∂xi =∑

(∂φ/∂xj)(∂xj/∂xi leads to the idea of covariant vectors Aj ∼ ∂φ/∂xj trans-forming via Ai =

∑(∂xj/∂xi)Aj (i.e. ∂/∂xi ∼ (∂xj/∂xj)∂/∂xj). Now define

connection coefficients or Christoffel symbols via (strictly one writes T γα = gαβT γβ

and T γα = gαβT βγ which are generally different - we use that notation here but

it is sometimes not used later when it is unnecessary due to symmetries, etc.)

(1.1) Γrki = −

r

k i

= −1

2

∑(∂igk + ∂kgi − ∂gik)gr = Γr

ik

91

92 3. GRAVITY AND THE QUANTUM POTENTIAL

(note this differs by a minus sign from some other authors). Note also that (1.1)follows from equations

(1.2) ∂gik + grkΓri + girΓr

k = 0

and cyclic permutation; the basic definition of Γimj is found in the transplantation

law dξi = Γimjdxmξj . Next for tensors Tα

βγ define derivatives Tαβγ|k = ∂kTα

βγ and

(1.3) Tαβγ|| = ∂T

αβγ − Γα

sTsβγ + Γs

βTαsγ + Γs

γTαβs

In particular covariant derivatives for contravariant and covariant vectors respec-tively are defined via

(1.4) ξi||k = ∂kξi − Γi

kξ = ∇kξi; ηm|| = ∂ηm + Γr

mηr = ∇ηm

respectively. Now to describe Weyl geometry one notes first that for Riemanniangeometry transplantation holds along with

(1.5) 2 = ‖ξ‖2 = gαβξαξβ ; ξαηα = gαβξαηβ

For Weyl geometry however one does not demand conservation of lengths andscalar products under affine transplantation as above. Thus assume d = (φβdxβ)where the covariant vector φβ plays a role analogous to Γα

βγ and one obtains

(1.6) d2 = 22(φβdxβ) = d(gαβξαξβ) =

= gαβ|γξαξβdxγ + gαβΓαργξρξβdxγ + gαβΓβ

ργξαξρdxγ

Rearranging etc. and using (1.5) again gives

(1.7) (gαβ|γ − 2gαβφγ) + gσβΓσαγ + gσαΓσ

βγ = 0;

Γαβγ = −

α

β γ

+ gσα[gσβφγ + gσγφβ − gβγφσ]

Thus we can prescribe the metric gαβ and the covariant vector field φγ and de-termine by (1.7) the field of connection coefficients Γα

βγ which admits the affinetransplantation law as above. If one takes φγ = 0 the Weyl geometry reduces toRiemannian geometry. This leads one to consider new metric tensors via a metricchange gαβ = f(xλ)gαβ and it turns out that (1/2)∂log(f)/∂xλ plays the role ofφλ. Here the metric change is called a gauge transformation and the ordinary con-

nections

αβ γ

constructed from gαβ are equal to the more general connections

Γαβγ constructed according to (1.7) from gαβ and φλ = (1/2)∂log(f)/∂xλ. The

generalized differential geometry is conformal in that the ratio

(1.8)ξαηα

‖ξ‖‖η‖ =gαβξαηβ

[(gαβξαξβ)(gαβηαηβ)]1/2

does not change under the gauge transformation gαβ → f(xλ)gαβ . Again if onehas a Weyl geometry characterized by gαβ and φα with connections determinedby (1.7) one may replace the geometric quantities by use of a scalar field f with

(1.9) gαβ = f(xλ)gαβ , φα = φα + (1/2)(log(f)|α; Γαβγ = Γα

βγ

without changing the intrinsic geometric properties of vector fields; the only changeis that of local lengths of a vector via 2 = f(xλ)2. Note that one can reduce

1. INTRODUCTION 93

φα to the zero vector field if and only if φα is a gradient field, namely Fαβ =φα|β − φβ|α = 0 (i.e. φα = (1/2)∂alog(f) ≡ ∂βφα = ∂αφβ). In this case one haslength preservation after transplantation around an arbitrary closed curve and thevanishing of Fαβ guarantees a choice of metric in which the Weyl geometry becomesRiemannian; thus Fαβ is an intrinsic geometric quantity for Weyl geometry; noteFαβ = −Fβα and

(1.10) Fαβ|γ = 0; Fµν|λ = Fµν|λ + Fλµ|ν + Fνλ|µ

Similarly the concept of covariant differentiation depends only on the idea of vectortransplantation. Indeed one can define covariant derivatives via

(1.11) ξα||β = ξα

|β − Γαβγξγ

In Riemannian geometry the curvature tensor is

(1.12) ξα||β|γ − ξα

||γ|β = Rαηβγξη; Rα

βγδ = −Γαβγ|δ + Γα

βδ|γ + ΓατδΓ

τβγ − Γα

τγΓτβδ

Using (1.8) one then can express this in terms of gαβ and φα but this is complicated.Equations for Rβδ = Rα

βαδ and R = gβδRβδ are however given in [12]. One notesthat in Weyl geometry if a vector ξα is given, independent of the metric, thenξα = gαβξβ will depend on the metric and under a gauge transformation one hasξα = f(xλ)ξα. Hence the covariant form of a gauge invariant contravariant vectorbecomes gauge dependent and one says that a tensor is of weight n if, under agauge transformation, Tα···

β··· = f(xλ)nTα···β··· . Note φα plays a singular role in (1.9)

and has no weight. Similarly√−g = f2√−g (weight 2) and Fαβ = gαµgβνFµν has

weight −2 while Fαβ = Fαβ√−g has weight 0 and is gauge invariant. SimilarlyFαβFαβ√−g is gauge invariant. Now for Weyl’s theory of electromagnetism onewants to interpret φα as an EM potential and one has automatically the Maxwellequations

(1.13) Fαβ|γ = 0; Fαβ|β = s

α

(the latter equation being gauge invariant source equations). These equations aregauge invariant as a natural consequence of the geometric interpretation of theEM field. For the interaction between the EM and gravitational fields one sets upsome field equations as indicated in [12] and the interaction between the metricquantities and the EM fields is exhibited there (there is much more on EM theorylater and see also Section 2.1.1).

REMARK 3.1.1. As indicated earlier in [12] Rijk is defined with a minus

sign compared with e.g. [723, 998] for example. There is also a difference indefinition of the Ricci tensor which is taken to be Gβδ = Rβδ − (1/2)gβδR in [12]with R = Rδ

δ so that Gµγ = gµβgγδGβδ = Rµγ − (1/2)gµγR with Gγ

η = Rηη−2R ⇒

Gηη = −R (recall n = 4). In [723] the Ricci tensor is simply Rβµ = Rα

βµα whereRα

βµν is the Riemann curvature tensor and R = Rηη again. This is similar to [998]

where the Ricci tensor is defined as ρj = Riji. To clarify all this we note that

(1.14) Rηγ = Rαηαγ = gαβRβηαγ = −gαβRβηγα = −Rα

ηγα

which reveals the minus sign difference.


2. SKETCH OF DEBROGLIE-BOHM-WEYL THEORY

From Chapters 1 and 2 we know something about Bohmian mechanics andthe quantum potential and we go now to the papers [869, 870, 871, 872, 873,874, 875, 876] by A. and F. Shojai to begin the present discussion (cf. also[8, 117, 118, 284, 668, 669, 831, 832, 834, 835, 836, 837, 838, 864,865, 866, 867, 868, 881] for related work from the Tehran school and [189,219, 611, 731, 840, 841, 872] for linking of dBB theory with Weyl geome-try). In nonrelativistic deBroglie-Bohm theory the quantum potential is Q =−(2/2m)(∇2|Ψ|/|Ψ|). The particles trajectory can be derived from Newton’s lawof motion in which the quantum force −∇Q is present in addition to the classicalforce −∇V . The enigmatic quantum behavior is attributed here to the quantumforce or quantum potential (with Ψ determining a “pilot wave” which guides theparticle motion). Setting Ψ =

√ρexp[iS/] one has

(2.1)∂S

∂t+|∇S|22m

+ V + Q = 0;∂ρ

∂t+∇ ·

(ρ∇S

m

)= 0

The first equation in (2.1) is a Hamilton-Jacobi (HJ) equation which is identicalto Newton’s law and represents an energy condition E = (|p|2/2m) + V + Q(recall from HJ theory −(∂S/∂t) = E(= H) and ∇S = p. The second equationrepresents a continuity equation for a hypothetical ensemble related to the particlein question. For the relativistic extension one could simply try to generalize therelativistic energy equation ηµνPµP ν = m2c2 to the form

(2.2) ηµνPµP ν = m2c2(1 +Q) =M2c2; Q = (2/m2c2)(|Ψ|/|Ψ|)

(2.3) M2 = m2

(1 + α

|Ψ||Ψ|

); α =

2

m2c2

This could be derived e.g. by setting Ψ =√

ρexp(iS/) in the Klein-Gordon (KG)equation and separating the real and imaginary parts, leading to the relativistic HJequation ηµν∂µS∂νS = M2c2 (as in (2.1) - note Pµ = −∂µS) and the continuityequation is ∂µ(ρ∂µS) = 0. The problem of M2 not being positive definite here(i.e. tachyons) is serious however and in fact (2.2) is not the correct equation (seee.g. [871, 873, 876]). One must use the covariant derivatives ∇µ in place of ∂µ

and for spin zero in a curved background there results (Q as above)

(2.4) ∇µ(ρ∇µS) = 0; gµν∇µS∇νS = M2c2; M

2 = m2eQ

To see this one must require that a correct relativistic equation of motion shouldnot only be Poincare invariant but also it should have the correct nonrelativisticlimit. Thus for a relativistic particle of mass M (which is a Lorentz invariantquantity) A =

∫dλ(1/2)M(r)(drµ/dλ)(drν/dλ) is the action functional where λ

is any scalar parameter parametrizing the path rµ(λ) (it could e.g. be the propertime τ). Varying the path via rµ → r′µ = rµ + εµ one gets (cf. [871])

(2.5) A→ A′ = A + δA = A +

∫dλ

[M

drµ

dλ

dεµ

dλ+

12

drµ drµ

dλεν∂ν

M

]dλ

2. SKETCH OF DEBROGLIE-BOHM-WEYL THEORY 95

By least action the correct path satisfies δA = 0 with fixed boundaries so theequation of motion is

(2.6) (d/dλ)(Muµ) = (1/2)uνuν∂µM;

M(duµ/dλ) = ((1/2)ηµνuαuα − uµuν)∂νM

where uµ = drµ/dλ. Now look at the symmetries of the action functional viaλ → λ+δ. The conserved current is then the Hamiltonian H = −L+uµ(∂L/∂uµ) =(1/2)Muµuµ = E. This can be seen by setting δA = 0 where

(2.7) 0 = δA = A′ − A =

∫dλ

[12uµuµuν∂νM + Muµ

duµ

dλ

]δ

which means that the integrand is zero, i.e. (d/dλ)[(1/2)Muµuµ] = 0. Since theproper time is defined as c2dτ2 = drµdrµ this leads to (dτ/dλ) =

√(2E/Mc2)

and the equation of motion becomes

(2.8) M(dvµ/dτ) = (1/2)(c2ηµν − vµvν)∂νM

where vµ = drµ/dτ . The nonrelativistic limit can be derived by letting the particlesvelocity be ignorable with respect to light velocity. In this limit the proper time isidentical to the time coordinate τ = t and the result is that the µ = 0 componentis satisfied identically via (r ∼ r)

(2.9) Md2r

dt2= −1

2c2∇M ⇒ m

(d2r

dt2

)= −∇

[mc2

2log

(M

µ

)]where µ is an arbitrary mass scale. In order to have the correct limit the termin parenthesis on the right side should be equal to the quantum potential so(mc2/2)log(M/µ) = (2/2m)(∇2|ψ|/|ψ|) and hence

(2.10) M = µexp[−(2/m2c2)(∇2|Ψ|/|Ψ|)]One infers that the relativistic quantum mass field is M = µexp[(2/2m)(|Ψ|/|Ψ|)](manifestly invariant) and setting µ = m we get (cf. also (2.12) below)

(2.11) M = mexp[(2/m2c2)(|Ψ|/|Ψ|)]If one starts with the standard relativistic theory and goes to the nonrelativisticlimit one does not get the correct nonrelativistic equations; this is a result of animproper decomposition of the wave function into its phase and norm in the KGequation (cf. also [110] for related procedures). One notes here also that (2.11)leads to a positive definite mass squared. Also from [871] this can be extendedto a many particle version and to a curved spacetime. However, for a particle ina curved background we will take (cf. [873] which we follow for the rest of thissection)

(2.12) ∇µ(ρ∇µS) = 0; gµν∇µS∇νS = M2c2; M

2 = m2eQ; Q =2

m2c2

g|Ψ||Ψ|

((2.11) suggests that M2 = m2exp(2Q) but (2.12) is used for compatibility withthe KG approach, etc., where exp(Q) ∼ 1 + Q - cf. remarks after (2.28) below- in any case the qualitative features are close here for either formula). Since,following deBroglie, the quantum HJ equation (QHJE) in (2.12) can be written in


the form (m2/M2)gµν∇µS∇νS = m2c2, the quantum effects are identical toa change of spacetime metric

(2.13) gµν → gµν = (M2/m2)gµν

which is a conformal transformation. The QHJE becomes then gµν∇µS∇νS =m2c2 where ∇µ represents covariant differentiation with respect to the metric gµν

and the continuity equation is then gµν∇µ(ρ∇νS) = 0. The important conclusionhere is that the presence of the quantum potential is equivalent to a curved space-time with its metric given by (2.13). This is a geometrization of the quantumaspects of matter and it seems that there is a dual aspect to the role of geometryin physics. The spacetime geometry sometimes looks like “gravity” and sometimesreveals quantum behavior. The curvature due to the quantum potential may havea large influence on the classical contribution to the curvature of spacetime. Theparticle trajectory can now be derived from the guidance relation via differentia-tion of (2.12) leading to the Newton equations of motion

(2.14) Md2xµ

dτ2+ MΓµ

νκuνuκ = (c2gµν − uµuν)∇νM

Using the conformal transformation above (2.14) reduces to the standard geodesicequation.

Now a general “canonical” relativistic system consisting of gravity and classicalmatter (no quantum effects) is determined by the action

(2.15) A =12κ

∫d4x√−gR+

∫d4x√−g

2

2m

(ρ

2DµSDµS − m2

2ρ

)where κ = 8πG and c = 1 for convenience. It was seen above that via deBrogliethe introduction of a quantum potential is equivalent to introducing a conformalfactor Ω2 = M2/m2 in the metric. Hence in order to introduce quantum effectsof matter into the action (2.15) one uses this conformal transformation to get(1 + Q ∼ exp(Q))

(2.16) A =12κ

∫d4x√−g(RΩ2 − 6∇µΩ∇µΩ)+

+∫

d4x√−g

( ρ

mΩ2∇µS∇µS −mρΩ4

)+∫

d4x√−gλ

[Ω2 −

(1 +

2

m2

√

ρ√

ρ

)]where a bar over any quantity means that it corresponds to the nonquantumregime. Here only the first two terms of the expansion of M2 = m2exp(Q) in(2.12) have been used, namely M2 ∼ m2(1 + Q). No physical change is involvedin considering all the terms. λ is a Lagrange multiplier introduced to identifythe conformal factor with its Bohmian value. One uses here gµν to raise of lowerindices and to evaluate the covariant derivatives; the physical metric (containingthe quantum effects of matter) is gµν = Ω2gµν . By variation of the action withrespect to gµν , Ω, ρ, S, and λ one arrives at the following quantum equations ofmotion:


(1) The equation of motion for Ω

(2.17) RΩ + 6Ω +2κ

mρΩ(∇µS∇µS − 2m2Ω2) + 2κλΩ = 0

(2) The continuity equation for particles ∇µ(ρΩ2∇µS) = 0(3) The equations of motion for particles (here a′ ≡ a)

(2.18) (∇µS∇µS −m2Ω2)Ω2√ρ +2

2m

[′

(λ√

ρ

)− λ

′√ρ

ρ

]= 0

(4) The modified Einstein equations for gµν

(2.19)Ω2

[Rµν − 1

2 gµνR]− [gµν′ − ∇µ∇ν ]Ω2 − 6∇µΩ∇νΩ + 3gµν∇αΩ∇αΩ+

+2κ

mρΩ2∇µS∇νS − κ

mρΩ2gµν∇αS∇αS + κmρΩ4gµν+

+κ2

m2

[∇µ√

ρ∇ν

(λ√

ρ

)+ ∇ν

√ρ∇µ

(λ√

ρ

)]− κ2

m2gµν∇α

[λ∇α√ρ√

ρ

]= 0

(5) The constraint equation Ω2 = 1 + (2/m2)[(√

ρ)/√

ρ]Thus the back reaction effects of the quantum factor on the background metric arecontained in these highly coupled equations (cf. also [27]). A simpler form of (2.17)can be obtained by taking the trace of (2.19) and using (2.17) which producesλ = (2/m2)∇µ[λ(∇µ√ρ)/

√ρ]. A solution of this via perturbation methods using

the small parameter α = 2/m2 yields the trivial solution λ = 0 so the aboveequations reduce to

(2.20) ∇µ(ρΩ2∇µS) = 0; ∇µS∇µS = m2Ω2; Gµν = −κT(m)µν − κT

(Ω)µν

where T(m)µν is the matter energy-momentum (EM) tensor and

(2.21) κT(Ω)µν =

[gµν−∇µ∇ν ]Ω2

Ω2+ 6

∇µΩ∇νΩω2

− 2gµν∇αΩ∇αΩ

Ω2

with Ω2 = 1+α(√

ρ/√

ρ). Note that the second relation in (2.20) is the Bohmianequation of motion and written in terms of gµν it becomes ∇µS∇µS = m2c2.

In the preceeding one has tacitly assumed that there is an ensemble of quan-tum particles so what about a single particle? One translates now the quantumpotential into purely geometrical terms without reference to matter parameters sothat the original form of the quantum potential can only be deduced after usingthe field equations. Thus the theory will work for a single particle or an ensembleand in this connection we make

REMARK 3.2.1. One notes that the use of ψψ∗ automatically suggests orinvolves an ensemble if it is to be interpreted as a probability density. Thus theidea that a particle has only a probability of being at or near x seems to mean thatsome paths take it there but others don’t and this is consistent with Feynman’suse of path integrals for example. This seems also to say that there is no suchthing as a particle, only a collection of versions or cloud connected to the particle


idea. Bohmian theory on the other hand for a fixed energy gives a one parameterfamily of trajectories associated to ψ (see here Section 2.2 and [197] for details).This is because the trajectory arises from a third order differential while fixingthe solution ψ of the second order stationary Schrodinger equation involves onlytwo “boundary” conditions. As was shown in [197] this automatically generatesa Heisenberg inequality ∆x∆p ≥ c; i.e. the uncertainty is built in when usingthe wave function ψ and amazingly can be expressed by the operator theoreticalframework of quantum mechanics. Thus a one parameter family of paths can beassociated with the use of ψψ∗ and this generates the cloud or ensemble auto-matically associated with the use of ψ. In fact, based on Remark 2.2.2, one mightconjecture that upon using a wave function discription of quantum particle motion,one opens the door to a cloud of particles, all of whose motions are incompletelygoverned by the SE, since one determining condition for particle motion is ignored.Thus automatically the quantum potential will give rise to a force acting on anysuch particular trajectory and the “ensemble” idea naturally applies to a cloud ofidentical particles (cf. also Theorem 1.2.1 and Corollary 1.2.1).

Now first ignore gravity and look at the geometrical properties of the confor-mal factor given via

(2.22) gµν = e4Σηµν ; e4Σ =M2

m2= exp

(α

η√

ρ√

ρ

)= exp

(α

η

√|T|√|T|

)where T is the trace of the EM tensor and is substituted for ρ (true for dust). TheEinstein tensor for this metric is

(2.23) Gµν = 4gµνηexp(−Σ) + 2exp(−2Σ)∂µ∂νexp(2Σ)

Hence as an Ansatz one can suppose that in the presence of gravitational effectsthe field equation would have a form

(2.24) Rµν −12Rgµν = κTµν + 4gµνeΣe−Σ + 2e−2Σ∇µ∇νe2Σ

This is written in a manner such that in the limit Tµν → 0 one will obtain (2.22).Taking the trace of the last equation one gets −R = κT− 12Σ + 24(∇Σ)2 whichhas the iterative solution κT = −R+ 12α[(

√R)/

√R] leading to

(2.25) Σ = α[(√|T|/

√|T|)] α[(

√|R|)/

√|R|)]

to first order in α.

One goes now to the field equations for a toy model. First from the aboveone sees that T can be replaced by R in the expression for the quantum potentialor for the conformal factor of the metric. This is important since the explicitreference to ensemble density is removed and the theory works for a single particleor an ensemble. So from (2.24) for a toy quantum gravity theory one assumes thefollowing field equations

(2.26) Gµν − κTµν − Zµναβexp(α

2Φ)∇α∇βexp

(−α

2Φ)

= 0


where Zµναβ = 2[gµνgαβ − gµαgνβ ] and Φ = (√|R|/

√|R|). The number 2 and

the minus sign of the second term are chosen so that the energy equation derivedlater will be correct. Note that the trace of (2.26) is

(2.27) R+ κT + 6exp(αΦ/2)exp(−αΦ/2) = 0

and this represents the connection of the Ricci scalar curvature of space timeand the trace of the matter EM tensor. If a perturbative solution is admit-ted one can expand in powers of α to find R(0) = −κT and R(1) = −κT −6exp(αΦ0/2)exp(−αΦ0/2) where Φ(0) =

√|T|/

√|T|. The energy relation can

be obtained by taking the four divergence of the field equations and since thedivergence of the Einstein tensor is zero one obtains

(2.28) κ∇νTµν = αRµν∇νΦ− α2

4∇µ(∇Φ)2 +

α2

2∇µΦΦ

For a dust with Tµν = ρuµuν and uµ the velocity field, the conservation of masslaw is ∇ν(ρMuν) = 0 so one gets to first order in α ∇µM/M = −(α/2)∇µΦor M2 = m2exp(−αΦ) where m is an integration constant. This is the correctrelation of mass and quantum potential.

In [873] there is then some discussion about making the conformal factordynamical via a general scalar tensor action (cf. also [867]) and subsequently onemakes both the conformal factor and the quantum potential into dynamical fieldsand creates a scalar tensor theory with two scalar fields. Thus start with a generalaction

(2.29) A =∫

d4x√−g

[φR− ω

∇µφ∇µφ

φ− ∇µQ∇µQ

φ+ 2Λφ + Lm

]The cosmological constant generally has an interaction term with the scalar fieldand here one uses an ad hoc matter Lagrangian

(2.30) Lm =ρ

mφa∇µS∇µS −mρφb − Λ(1 + Q)c + αρ(eQ − 1)

(only the first two terms 1 + Q from exp(Q) are used for simplicity in the thirdterm). Here a, b, c are constants to be fixed later and the last term is chosen(heuristically) in such a manner as to have an interaction between the quantumpotential field and the ensemble density (via the equations of motion); further theinteraction is chosen so that it vanishes in the classical limit but this is ad hoc.Variation of the above action yields

(1) The scalar fields equation of motion

(2.31) R+2ω

φφ− ω

φ2∇µφ∇µφ + 2Λ+

+1φ2∇µQ∇µQ +

a

mρφa−1∇µS∇µS −mbρφb−1 = 0

(2) The quantum potential equations of motion

(2.32) (Q/φ)− (∇µQ∇µφ/φ2)− Λc(1 + Q)c−1 + αρexp(Q) = 0

(3) The generalized Einstein equations

(2.33) Gµν − Λgµν = − 1

φT

µν − 1φ

[∇µ∇ν − gµν]φ +ω

φ2∇µφ∇νφ−

− ω

2φ2gµν∇αφ∇αφ +

1φ2∇µQ∇νQ− 1

2φ2gµν∇αQ∇αQ

(4) The continuity equation ∇µ(ρφa∇µS) = 0(5) The quantum Hamilton Jacobi equation

(2.34) ∇µS∇µS = m2φb−a − αmφ−a(eQ − 1)

In (2.31) the scalar curvature and the term ∇µS∇µS can be eliminated using(2.33) and (2.34); further on using the matter Lagrangian and the definition of theEM tensor one has

(2.35) (2ω − 3)φ = (a + 1)ρα(eQ − 1)− 2Λ(1 + Q)c + 2Λφ− 2φ∇µQ∇µQ

(where b = a + 1). Solving (2.32) and (2.35) with a perturbation expansion in αone finds

(2.36) Q = Q0 + αQ1 + · · · ; φ = 1 + αQ1 + · · · ;√

ρ =√

ρ0 + α√

ρ1 + · · ·where the conformal factor is chosen to be unity at zeroth order so that as α → 0(2.34) goes to the classical HJ equation. Further since by (2.34) the quantummass is m2φ + · · · the first order term in φ is chosen to be Q1 (cf. (2.12)).Also we will see that Q1 ∼

√ρ/√

ρ plus corrections which is in accord withQ as a quantum potential field. In any case after some computation one ob-tains a = 2ωk, b = a + 1, and = (1/4)(2ωk + 1) = (1/4)(a + 1) = b/4 withQ0 = [1/c(2c − 3)][−(2ωk + 1)/2Λ]k

√ρ0 − (2c2 − c + 1) while ρ0 can be deter-

mined (cf. [873] for details). Thus heuristically the quantum potential can beregarded as a dynamical field and perturbatively one gets the correct dependenceof quantum potential upon density, modulo some corrective terms.

One goes next to a number of examples and we only consider here the confor-mally flat solution (cf. also [869]). Thus take gµν = exp(2Σ)ηµν where Σ << 1.One obtains from (2.24)

(2.37) Rµν = ηµνΣ + 2∂µ∂νΣ ⇒ Gµν = 2∂µ∂νΣ− 2ηµνΣ

One can solve this iteratively to get

(2.38) R(0) = −κT ⇒ Σ(0) = −κ

6−1

T;

R(1) = −κT + 3α√|T|√|T|

⇒ Σ(1) = −κ

6−1

T +α

2√|T|√|T|

Consequently

(2.39) Σ = −κ

6−1

T +α

2√|T|√|T|

+ · · ·

The first term is pure gravity, the second pure quantum, and the remaining termsinvolve gravity-quantum interactions. Other impressive examples are given (cf.also [869]).


One goes now to a generalized equivalence principle. The gravitational effectsdetermine the causal structure of spacetime as long as quantum effects give itsconformal structure. This does not mean that quantum effects have nothing todo with the causal structure; they can act on the causal structure through backreaction terms appearing in the metric field equations. The conformal factor of themetric is a function of the quantum potential and the mass of a relativistic particleis a field produced by quantum corrections to the classical mass. One has shownthat the presence of the quantum potential is equivalent to a conformal mapping ofthe metric. Thus in different conformally related frames one feels different quan-tum masses and different curvatures. In particular there are two frames with onecontaining the quantum mass field and the classical metric while the other containsthe classical mass and the quantum metric. In general frames both the spacetimemetric and the mass field have quantum properties so one can state that differentconformal frames are identical pictures of the gravitational and quantum phenom-ena. We feel different quantum forces in different conformal frames. The questionthen arises of whether the geometrization of quantum effects implies conformalinvariance just as gravitational effects imply general coordinate invariance. Onesees here that Weyl geometry provides additional degrees of freedom which can beidentified with quantum effects and seems to create a unified geometric frameworkfor understanding both gravitational and quantum forces. Some features here are:(i) Quantum effects appear independent of any preferred length scale. (ii) Thequantum mass of a particle is a field. (iii) The gravitational constant is also a fielddepending on the matter distribution via the quantum potential (cf. [867, 874]).(iv) A local variation of matter field distribution changes the quantum potentialacting on the geometry and alters it globally; the nonlocal character is forced bythe quantum potential (cf. [868]).

2.1. DIRAC-WEYL ACTION. Next (still following [873]) one goes toWeyl geometry based on the Weyl-Dirac action

(2.40) A =∫

d4x√−g(FµνFµν − β2 WR+ (σ + 6)β;µβ;µ + Lmatter)

Here Fµν is the curl of the Weyl 4-vector φµ, σ is an arbitrary constant and β is ascalar field of weight −1. The symbol “;” represents a covariant derivative undergeneral coordinate and conformal transformations (Weyl covariant derivative) de-fined as X;µ = W∇µX −NφµX where N is the Weyl weight of X. The equationsof motion are then

(2.41) Gµν = −8π

β2(Tµν + Mµν) +

2β

(gµνW∇αW∇αβ −W∇µW∇νβ)+

+1β2

(4∇µβ∇νβ − gµν∇αβ∇αβ) +σ

β2(β;µβ;ν − 1

2gµνβ;αβ;α);

W∇µFµν =12σ(β2φµ + β∇µβ) + 4πJµ;

R = −(σ + 6)W β

β+ σφαφα − σW∇αφα +

ψ

2β


where

(2.42) Mµν = (1/4π)[(1/4)gµνFαβFαβ − Fµα F να]

and

(2.43) 8πTµν =

1√−g

δ√−gLmatter

δgµν; 16πJµ =

δLmatter

δφµ; ψ =

δLmatter

δβ

For the equations of motion of matter and the trace of the EM tensor one usesinvariance of the action under coordinate and gauge transformations, leading to

(2.44) W∇νTµν − T

∇µβ

β= Jαφαµ −

(φµ +

∇µβ

β

)W∇αJα;

16πT− 16πW∇µJµ − βψ = 0

The first relation is a geometrical identity (Bianchi identity) and the second showsthe mutual dependence of the field equations. Note that in the Weyl-Dirac theorythe Weyl vector does not couple to spinors so φµ cannot be interpreted as theEM potential; the Weyl vector is used as part of the spacetime geometry andthe auxillary field (gauge field) β represents the quantum mass field. The gravityfields gµν and φµ and the quantum mass field determine the spacetime geometry.Now one constructs a Bohmian quantum gravity which is conformally invariantin the framework of Weyl geometry. If the model has mass this must be a field(since mass has non-zero Weyl weight). The Weyl-Dirac action is a general Weylinvariant action as above and for simplicity now assume the matter Lagrangiandoes not depend on the Weyl vector so that Jµ = 0. The equations of motion arethen

(2.45) Gµν = −8π

β2(Tµν + Mµν) +

2β

(gµνW∇αW∇αβ −W∇µW∇νβ)+

+1β2

(4∇µβ∇νβ − gµν∇αβ∇αβ) +σ

β2

(β;µβ;ν − 1

2gµνβ;αβ;α

);

W∇νFµν =12σ(β2φµ + β∇µβ); R = −(σ + 6)

W β

β+ σφαφα − σW∇αφα +

ψ

2β

The symmetry conditions are

(2.46) W∇νTµν − T(∇µβ/β) = 0; 16πT− βψ = 0

(recall T = Tµνµν). One notes that from (2.45) results W∇µ(β2φµ + β∇µβ) = 0 so

φµ is not independent of β. To see how this is related to the Bohmian quantumtheory one introduces a quantum mass field and shows it is proportional to theDirac field. Thus using (2.45) and (2.46) one has

(2.47) β +16βR =

4π

3T

β+ σβφαφα + 2(σ − 6)φγ∇γβ +

σ

β∇µβ∇µβ

This can be solved iteratively via

(2.48) β2 = (8πT/R)− 1/[(R/6)− σφαφα]ββ + · · ·


Now assuming Tµν = ρuµuν (dust with T = ρ) we multiply (2.46) by uµ and sumto get

(2.49) W∇ν(ρuν)− ρ(uµ∇µβ/β) = 0

Then put (2.46) into (2.49) which yields

(2.50) uνW∇νuµ = (1/β)(gµν − uµuν)∇νβ

To see this write (assuming gµν∇νβ = ∇µβ)

(2.51) W∇ν(ρuµuν) = uµW∇νρuµ + ρuνW∇νuµ ⇒

⇒ uµ

(uµ∇µβ

β

)+ uνW∇νuµ − ∇

µβ

β= 0 ⇒ uνW∇νuµ = (1− uµuµ)

∇µβ

β=

(gµν − uµuµgµν)∇νβ

β= (gµν − uµuν)

∇νβ

β

which is (2.49). Then from (2.48)

(2.52) β2(1) =8πT

R ; β2(2) =8πT

R

(1− 1

(R/6)− σφαφα

√

T√T

); · · ·

Comparing with (2.14) and (2.3) shows that we have the correct equations for theBohmian theory provided one identifies

(2.53) β ∼M;8πT

R ∼ m2;1

σφαφα − (R/6)∼ α =

2

m2c2

Thus β is the Bohmian quantum mass field and the coupling constant α (whichdepends on ) is also a field, related to geometrical properties of spacetime. Onenotes that the quantum effects and the length scale of the spacetime are related.To see this suppose one is in a gauge in which the Dirac field is constant; applya gauge transformation to change this to a general spacetime dependent function,i.e.

(2.54) β = β0 → β(x) = β0exp(−Ξ(x)); φµ → φµ + ∂µΞ

Thus the gauge in which the quantum mass is constant (and the quantum force iszero) and the gauge in which the quantum mass is spacetime dependent are relatedto one another via a scale change. In particular φµ in the two gauges differ by−∇µ(β/β0) and since φµ is a part of Weyl geometry and the Dirac field representsthe quantum mass one concludes that the quantum effects are geometrized (cf.also (2.45) which shows that φµ is not independent of β so the Weyl vector isdetermined by the quantum mass and thus the geometrical aspects of the manifoldare related to quantum effects).


2.2. REMARKS ON CONFORMAL GRAVITY. We go here to a se-ries of papers by Arias, Bonal, Cardenas, Gonzalez, Leyva, Martin, and Quiros(cf. [46, 47, 48, 133, 181, 182, 796, 797, 798, 799]) and sketch at some lengthsome results concerning Brans-Dicke theory, conformal gravity, and deBroglie-Bohm-Weyl (dBBW) theory (many other topics are also covered in these paperswhich we omit here - cf. also [24, 37, 801, 974, 975]). The presentation in[188] of this material is difficult to read and we try here for a smoother develop-ment. In [188] we started with [797, 799] and then gave a later reformulationfrom [798]; we expand upon this now (still in a more or less chronological order[799, 797, 798, 796]) and try to make matters clearer. Questions about the phys-ical significance of Riemannian geometry in relativity have been raised in the past(cf. [150, 301]) due to the arbitrariness in the metric tensor resulting from theindefiniteness in the choice of units of measure. In fact Brans-Dicke (BD) theorywith a changing dimensionless gravitational coupling constant Gm2 ∼ φ−1 (withm the intertial mass of some elementary particle and φ the BD field - = c = 1here) can be formulated in two different ways since either m or G could vary withposition in spacetime. The choice G ∼ φ−1 with m = const. leads to the Jordanframe (JF) formalism based on the Lagrangian

(2.55) LBD[g, φ] =√−g

16π

(φR− ω

φgnm∇nφ∇mφ

)+ LM [g]

where R is the curvature scalar, ω is the BD coupling constant, and LM [g] is theLagrangian density for ordinary matter minimally coupled to the scalar field. Onthe other hand the choice m ∼ φ−1/2 with G constant leads to the Einstein frame(EF) BD theory based on the Lagrangian

(2.56) LBD =√−g

16π

(R−

(ω +

32

)gnm∇nφ∇mφ

)+ LM [g, φ]

where now in the EF metric g the ordinary matter is nonminimally coupled to thescalar field φ ≡ log(φ) through the Lagrangian density LM [g, φ]. Both JF and EFformulations of BD gravity are equivalent representations of the same physical sit-uation since they both belong to the same conformal class (cf. [150]); in particularLBD

EF ≡ LBDJF via a rescaling of spacetime metric g → g = φg or gab = φgab where

φ is smooth and nonvanishing. This rescaling can be interpreted as a particulartransformation of the physical units and any dimensionless number (e.g. Gm2) isinvariant; experimental observations are unchanged since spacetime coincidencesare not affected. Hence both based formulations (one based on varying G andthe other on varying m are indistinguishable) and one has physically equivalentrepresentations of a same physical situation. The same line of reasoning can beapplied if minimal and nonminimal coupling to matter are interchanged via

(2.57) (A) LGR[g, φ] =√−g

16π

(φR− ω

φgnm∇nφ∇mφ

)+ LM [g, φ];

(B) LGR =√−g

16π

(R−

(ω +

32

)gnm∇nφ∇mφ

)+ LM [g]


Both Lagrangians represent equivalent pictures of GR and (B) is simply GR witha scalar field as an additional source of gravity (EFGR) and its conformally equiv-alent Lagrangian (A) refers to Jordan frame GR (JFGR). The field equationsderivable from Lagrangian (B) are

(2.58) Gab = 8πTab +(

ω +32

)(∇aφ∇bφ−

12gabg

nm∇nφ∇mφ

);

φ = 0; ∇nTna = 0; = gnm∇n∇m

where Gab = Rab − (1/2)gabR and Tab = (2/√

g)(∂/∂gab)(√−gLM ). Some disad-

vantages for JFGR historically involve first that the BD scalar field is nonminimallycoupled both to scalar curvature and to ordinary matter so the gravitational con-stant G varies as G ∼ φ−1. At the same time the material test particles don’tfollow the geodesics of the geometry since they are acted on by both the metric fieldand the scalar field. In particular masses vary from point to point in spacetimeso as to preserve a constant Gm2 (so m ∼ φ1/2). The most serious (but illusory)objection is linked with the formulation of the theory in unphysical variables sothat the kinetic energy of the scalar field is not positive deffinite (cf. [351]). How-ever one shows in [799] that the indefiniteness in the sign of the energy density inthe Jordan frame is only apparent; in fact once the scalar field energy density ispositive definite in the Einstein frame it is also in the Jordan frame.

Usually the JF formulation of BD gravity is linked with Riemannian geometry(cf. [150]). This is directly related to the fact that in the JFBD formalism ordi-nary matter is minimally coupled to the scalar BD field through LM [g] in (2.55).This means that point particles follow the geodesics of the Riemannian geometry.This geometry is based on the parallel transport law and length preservation law

(2.59) dξa = −γanmξmdxn; dg(ξ, ξ) = 0

where g(ξ, ξ) = gnmξnξm and γanm are the affine connections of the manifold.

These postulates mean that γabc = Γa

bc = (1/2)gan(gnb,c + gnc,b− gbc,n) (Christoffelsymbols). After the rescaling gab = φgab the above parallel transport and lengthrules become (recall φ ∼ log(φ))

(2.60) dξa = −γanmξmdxn; dg(ξ, ξ) = dxn∇nφg(ξ, ξ);

γabc = Γa

bc −12(∇bφδa

c + ∇cφδab − ∇aφgbc)

Thus the affine connections of the manifold don’t coincide with the Christoffel sym-bols of the metric and one has a Weyl type manifold. Thus JF and EF Lagrangiansof both BD and GR theories are connected by conformal rescaling of the metrictogether with scalar field redefinition. This means JF and EF formulations on theone hand and Riemannian and Weyl type geometries on the other form conformalequivalence classes (uniquely defined only after the coupling of matter fields to themetric). In BD theory for example matter minimally couples to the JF so the testparticles follow the geodesics of the Riemannian geometry (i.e. JFBD is linkedto Riemannian geometry) while EFBD theory (conformal to JFBD) is linked toa Weyl type geometry. Similarly EFGR is linked with Riemannian geometry andJFGR (conformal to EFGR) is linked to a Weyl type geometry. When the matter


part of the Lagrangian is absent both BD and GR theories can be interprted onthe grounds of either Riemann or Weyl type geometry and one can conclude thatBD and GR theory with an extra scalar field coincide.

The field equations of JFGR can be derived, either directly from (2.57) (equa-tion (A)) or by conformally mapping (2.58) back to the JF metric according tog = φg, to obtain

(2.61) φ = 0; ∇nTna =12φ−1∇aφT ;

Gab =8π

φTab +

ω

φ2(∇aφ∇bφ−

12gabg

nm∇− nφ∇mφ) +1φ

(∇a∇bφ− gabφ)

where Tab = (2/√−g)(∂/∂gab)(

√−gLM ) is the stress energy tensor for ordinarymatter in the Jordan frame. The energy is not conserved because the scalar fieldφ exchanges energy with the metric and with the matter fields. The equation ofmotion of an uncharged spinless mass point that is acted both by the JF metricfield g and the scalar field φ is

(2.62)d2xa

ds2= −Γa

nm

dxm

ds

dxn

ds− 1

2φ−1∇nφ

(dxn

ds

dxa

ds− gan

)This does not coincide with the geodesic equation of the JF metric and this,together with the more complex structure of (2.61) in comparison to (2.58), in-troduces additional complications in the dynamics of matter fields. The fact thatthe Jordan frame does not lead to a well defined energy momentum tensor for thescalar field is perhaps the most serious objection to this representation (cf. [351]).Thus the kinetic energy of the JF scalar field is negative definite or indefinite un-like the Einstein frame where for ω > −(3/2) it is positive definite; this impliesno stable ground state and hence unphysical variables (cf. [351]). However al-though the right side of (2.61) does not have a definite sign the scalar field stressenergy tensor can be given the canonical form (cf. [842] for example). In [799]one obtains the same result as in [842] by rewriting equation (2.61) in terms ofaffine magnitudes in the Weyl type manifold. Thus the affine connections of theJF (Weyl type) manifold γa

bc are related with the Christoffel symbols of the JFmetric through γa

bc = Γabc +(1/2)φ−1(∇bφδa

c +∇cφδab −∇aφgbc) and one can define

the affine Einstein tensor γGab via the γabc instead of the Christoffel symbols of Γa

bc

so that (2.61) becomes

(2.63) γGab =8π

φTab +

[ω + (3/2)]φ2

(∇aφ∇bφ− (1/2)gabgnm∇− nφ∇mφ)

Now (φ/8π) times the second term in the right side has the canonical form for thestress energy tensor (true stress energy tensor) while (φ/8π) times the sum of thesecond and third terms in the right side will be called the effective stress energytensor for the BD scalar field φ (cf. (2.58)). Thus once the scalar field energydensity is positive definite in the Einstein frame it is also in the Jordan frame.This removes the main physical objection to the Jordan frame formulation of GR.

Another remarkable feature of JFGR is the invariance under the following


conformal transformations

(2.64) (A) gab = φ2gab; φ = φ−1 (B) gab = fgab; φ = f−1φ

where f is smooth. Also JFBD based on (A) in (2.57) is invariant under the moregeneral rescaling (cf. [350, 796])

(2.65) gab = φ2αgab; φ = φ1−2α; ω =ω − 6α(α− 1)

(1− 2α)2

for α = 1/2. The conformal invariance of a given theory of gravitation under atransformation of physical units is very desirable and in particular (A) in (2.57)is thus a better candidate for BD theory than classical theories given by (2.55),(2.56), or (B) in (2.57) which are not invariant.

We go now to [798] where a number of argments from [797, 799] are re-peated and amplified for greater clarity. It has been demonstrated already thatGR with an extra scalar field and its conformal formulation (JFGR) are differ-ent but physically equivalent representations of the same theory. The claim isbased on the argument that spacetime coincidences (coordinates) are not affectedby a conformal rescaling of the spacetime metric () gab = Ω2gab where Ω2 isa smooth nonvanishing function on the manifold. Thus the experimental obser-vations (measuements) being nothing but verifications of these coincidences areunchanged too by (). This means that canonical GR and its conformal imageare experimentally indistinguishable. Now a possible objection to this claim couldbe based on the following argument (which will be refuted). ARGUMENT: Incanonical GR the matter fields couple minimally to the metric g that determinesmetrical relations on a Riemannian spacetime. Hence matter particles follow thegeodesics of the metric g (in Riemannian geometry) and their masses are constantover the spacetime manifold, i.e. it is the metric which matter “feels” or perhapsthe “physical metric”. Under the conformal rescaling the matter fields becomenon-minimally coupled to the conformal metric g and hence matter particles donot follow the geodesics of this last metric. Furthermore, it is not the metric thatdetermines metrical relations on the Riemannian manifold. This line of reasoningleads to the following conclusion. Although canonical GR and its conformal im-age may be physically equivalent theories, nevertheless, the physical metric is thatwhich determines metrical relations on a Riemannian spacetime and the conformalmetric g is not the physical metric. REFUTATION - to be developed: Un-der the conformal rescaling () not only the Lagrangian of the theory is mappedinto its conformal Lagrangian but the spacetime geometry itself is mapped toointo a conformal geometry. In this last geometry metrical relations involve boththe conformal metric g and the conformal factor Ω2 generating (). Hence inthe conformal Lagrangian the matter fields should “feel” both the metric and thescalar function Ω, i.e. the matter particles would not follow the geodesics of theconformal metric alone. The result is that under () the “physical” metric of theuntransformed geometry is effectively mapped into the “physical” metric of theconformal geometry. This “missing detail” has apparently been a source of longstanding confusion and, although details have been sketched already, more will beprovided.


Another question regarding metric theories of spacetime is also clarified, namelythe physical content of a given theory of spacetime should be contained in the in-variants of the group of position dependent transformations of units and coordinatetransformations (cf. [150]). All known metric theories of spacetime, including GR,BD, and scalar-tensor theories in general fufill the requirement of invariance underthe group of coordinate transformations. It is also evident that any consistent for-mulation of a given effective theory of spacetime must be invariant also under thegroup of transformations of units of length, time, and mass. This aspect is treatedbelow and one shows that the only consistent formulation of gravity (among thosestudied here) is the conformal representation of GR.

Now with some repetition one considers various Lagrangians again. First isthat for GR with an extra scalar field, namely (α ≥ 0 is a free parameter)

(2.66) LGR =√−g(R− α(∇φ)2 + 16π

√−gLM

(note (∇φ)2 = gmnφ,mφ,n). When φ is constant or α = 0 one recovers the usualEinstein theory. Under the conformal rescaling () with Ω2 = exp(φ) the La-grangian in (2.66) is mapped into its conformal Lagrangian (cf. (2.55))

(2.67) LGR =√−geφ(R− (α− (3/2))(∇φ)2 + 16π

√−ge2φLM ≡

≡√−g

(φR−

(α− 3

2

)(∇φ)2

φ

)+ 16π

√−gLM

(the latter expression having a more usual BD form). Due to the minimal cou-pling of the scalar field φ to the curvature in canonical GR ((2.66)) the effectivegravitational constant G (set equal to 1 in (2.66)) is a real constant. The minimalcoupling of the matter fields to the metric in (2.66) entails that matter particlesfollow the geodesics of the metric g. Hence the inertial mass m is constant tooover spacetime. This implies that the dimensionless gravitational coupling con-stant Gm2 (c = = 1) is constant in spacetime - unlike BD theory where thisevolves as φ−1. This is a conformal invariant feature of GR since dimensionless con-stants do not change under (); in other words in conformal GR Gm2 is constantas well. However in this case ((2.67)) the effective gravitational constant varieslike G ∼ φ−1 and hence the particle masses vary like m = exp(φ/2)m = φ1/2m.According to [301] the conformal transformation () (with Ω2 = φ) can be in-terpreted as a transformation of the units of length time and reciprocal mass; inparticular there results ds = φ−1/2ds while m−1 = φ−1/2m−1. A careful look at(2.66) - (2.67) shows that Einstein’s laws of gravity derivable from (2.66) changeunder the units transformation () and this seems to be a serious drawback ofcanonical GR (and BD theory and scalar-tensor theories in general) since in anyconsistent theory of spacetime the laws of physics must be invariant under a changeof the units of length, time, and mass. This will be clarified below where it is shownthat () with Ω2 = φ = exp(φ) cannot be taken properly as a units transforma-tion. It is just a transformation allowing jumping from one formulation to itsconformal equivalent.


In [797, 799] one claimed that canonical GR ((2.66)) and its conformal La-grangian (2.67) are physically equivalent theories since they are indistinguishablefrom the observational point of view. However it is common to believe that onlyone of the conformally related metrics is the “physical” metric, i.e. that whichdetrmines metrical relations on the spacetime manifold. The reasoning leading tothis conclusion is based on the following analysis. Take for instance GR with anextra scalar field. Due to the minimal coupling of the matter fields to the metricin (2.66) the matter particles follow the geodesics of the metric g, namely

(2.68)d2xa

ds2+ Γa

mn

dxm

ds

dxn

ds= 0

where Γabc = (1/2)gan(gbn,c + gcn,b − gbc,n) are the Christoffel symbols of the

metric g. These coincide with the geodesics of the Riemannian geometry wheremetrical relations are given by g via g(X, Y ) = gmnXmY n and the line elementds2 = gmndxmdxn, etc. It is the reason why canonical GR based on (2.66) isnaturally linked with Riemannian geometry (it is the same for JFBD since thematter fields couple minimally to the spacetime metric). The units of this geom-etry ar constant over the manifold. On the other hand since the matter fields arenon-minimally coupled to the metric in the conformal GR the matter particleswould not follow the geodesics of the conformal metric g but rather curves whichare solutions of the equation conformal to (2.68), namely (2.62) where now Γa

bc arethe Christoffel symbols of the metric g conformal to g. Hence if one assumes thatthe spacetime geometry is fixed to be Riemannian and that it is unchanged underthe conformal rescaling () with Ω2 = φ one efffectively arrives at the conclusionthat g is the “physical” metric. However this assumption is wrong and is thesource of much long standing confusion (to be further clarified below).

REMARK 3.2.2. One notes that conformal Riemannian geometry (corre-sponding to Weyl geometry here) develops as follows. Let λ(t) be a curve withlocal coordinates xa(t) and let X with local coordinates Xa = dxa/dt be a tangentvector to λ(t). The covariant derivative of a given vector field Y along λ is givenby

(2.69)DY a

∂t=

∂Y a

∂t+ γa

mnY m dxn

dt

where γab is a symmetric connection. Given a metric g on a manifold M the

Riemannian geometry is fixed by the condition that there is a unique torsion free(symmetric) connction on M such that the covariant derivative of g is zero; thenparallel transport of vectors Y (DY a/∂t) = 0 and this preserves scalar products,i.e. dg(Y , Y ) = 0. The laws of parallel transport and length preservation entailthat the symmetric connection γa

bc coincides with the Christoffel symbols of themetric g, so γa

bc = Γabc. Suppose now that Y transform under () (with Ω2 = φ)

as Y = h(φ)Y a; then dg(Y, Y ) = −d[log(φh2)]g(Y, Y ) which resembles the lawof length transport in Weyl geometry. Hence given a Riemannian geometry onM , under () with Ω2 = φ one arrives at a Weyl geometry on M. The parallel


transport law conformal to (2.69) is

(2.70)DY a

∂t+

∂

∂t(log(h)Y a;

DY a

∂t=

∂Y a

∂t+ γa

mnY m dxn

dt

Here γabc is the symmetric connection on the Weyl manifold M related to the

Christoffel symbols of the conformal metric g via

(2.71) γabc = Γa

bc +12φ−1(φ,bδ

ac + φ,cδ

ab − gbcg

anφ,n)

There will be particle motions as in (2.62) and in particular Weyl geometry includesunits of measure with point dependent length.

REMARK 3.2.3. We go now to transformations of units following [796,798]. Consider two Lagrangians

(2.72) L1 =√−g(R− α(∇φ)2); L2 =

√−g

(φR−

(α− 3

2

)(∇φ)2

φ

)with respect to transformations () (note L2 can be obtained from L1 by rescalingg → φg and φ → log(φ)). Consider first conformal transformations gab = φσgab

(σ arbitrary) leading to

(2.73) L1 =√−g(φσR + [(3σ(3/2)σ2]φ−2−σ − αφσ)(∇φ)2)

Hence the laws of gravity it describes change under this transformation; in partic-ular in the conformal (tilde) frame the effective gravitational constant depends onφ due to the nonminimal coupling between the scalar field and the curvature. Onthe other hand L2 is mapped to

(2.74) L2 =√−g

(φ1−σR− (α− (3/2)− 3σ + (3/2)σ2)

(1− σ)2φσ−1(∇φ1−σ

)Consequently if we introduce a new scalar field variable φ = φ1−σ and redefine thefree parameter of the theory via α = [α + 3σ(σ − 2)]/(1− σ)2 the Lagrangian L2

takes the form

(2.75) L2 =√−g

(φR−

(α− 3

2

)(∇φ)2

φ

)Hence the Lagrangian L2 is invariant in form under the conformal transformationand field transformation indicated. The composition of two such transformationswith parameters σ1 and σ2 yields a transformation of the same form with pa-rameter σ3 = σ1 + σ2 − σ1σ2. The identity transformation involves σ = 0 andthe inverse for σ is a transformation with parameter σ = −[σ/(1 − σ)]. Henceexcluding the value σ = 1 we have a one parameter Abelian group of transforma-tions (σ3(σ1, σ2) = σ3(σ2, σ1)). This all leads to the conclusion that, since anyconsistent theory of spacetime must be invariant under the one parameter groupof transformations of units (length, time, mass), spacetime theores based on theLagrangian for pure GR of the form L1 ar not consistent theories while those basedon Lagrangians of the form L2 may in principle be consistent formulations of aspacetime theory. In particular canonical GR and the Einstein frame formulationof BD theory are not consistent formulations.

3. THE SCHRODINGER EQUATION IN WEYL SPACE 111

Consider now, separately, matter Lagrangians

(2.76) (A)√−gφ2LM ; (B)

√−gLM

((B) involves minimal coupling of matter to the metric and (A) is non-minimal.Under gab = φσgab we have (A) →

√−gφ2−2σLM and hence via scalar field

redefinition (A) is invariant in form under the group of units transformations.However (B) with minimal coupling is not invariant and hence JFBD based onLBD = L2 + 16π

√−gLM coupling (L2 as in (2.72)) is not a consistent theory ofspacetime. The only surviving candidate is conformal GR based on (2.67), namelyL2 + 16π

√−gφ2LM , and this theory provides a consistent formulation of the lawsof gravity. Thus the conformal version of GR involving Weyl type geometry isthe object to study and this is picked up again in [133, 796] along with someconnections to Bohmian theory. Various other topics involving cosmology and sin-gularities are also studied in [796, 797, 798, 799] but we omit this here.

REMARK 3.2.4. We make a few comments now following [133, 796] aboutWeyl geometry and the quantum potential. First we have seen that Einstein’s GRis incomplete and a Weylian form seems preferable. Secondly there seems to beevidence that a Weylian form can solve (or smooth) various problems involvingsingularities (cf. [796, 797, 798, 799] for some information in this direction). Onerecalls also the arguments emanating from string theory that a dilaton should cou-pled to gravity in the low energy limit (cf. [429]). It is to be noted that Weylspacetimes conformally linked to Riemannian structure (such as conformal GR)are called Weyl integrable spacetimes (WISP). The terminology arises from thecondition gab;c = 0 for a Riemannian space (where the symbol “;” denotes covari-ant differentiation. Then if g = Ω2g with Ω2 = φ = exp(ψ) (note we are switchingthe roles of g and g used earlier) there results gab;c = ψ,cgab (affine covariantderivative involving Γa

bc) which are the Weyl affine connection coefficients of theconformal manifold. Comparing to the requirement gab;c = wcgab for Weyl geome-tries (wc is the Weyl gauge vector) we see that Weyl structures conformally linkedto Riemannian geometry have the property that wc = ψ,c (ψ here corresponds tothe dilaton) and this is the origin of the term integrable since via d = dxnψ,n

one arrives at∮

d = 0 for WISP (which eliminates the second clock effect oftenused to critisize Weyl spacetime). Note that the equations of motion of a free par-ticle (or geodesic curves) in the WIST are given by (2.62) (with φ = log(ψ)) andsetting e.g. exp(ψ) = 1 + Q where Q is the quantum potential one can regard thelast term in (2.62) as the quantum force (see here Section 3.2 for a more refinedapproach). In any event the moral here is that Weyl geometry implicitly containsthe quantum effects of matter - it is already a quantum geometry! In particular afree falling test particle would not “feel” any special quantum force since the effectis built into the free fall.

3. THE SCHRODINGER EQUATION IN WEYL SPACE

We go now to Santamato [840] and derive the SE from classical mechanics inWeyl space (i.e. from Weyl geometry - cf. also [63, 188, 189, 219, 224, 490,841, 989]). The idea is to relate the quantum force (arising from the quantum


potential) to geometrical properties of spacetime; the Klein-Gordon (KG) equationis also treated in this spirit in [219, 841] and we discuss this later. One wants toshow how geometry acts as a guidance field for matter (as in general relativity).Initial positions are assumed random (as in the Madelung approach) and thusthe theory is statistical and is really describing the motion of an ensemble. Thusassume that the particle motion is given by some random process qi(t, ω) in amanifold M (where ω is the sample space tag) whose probability density ρ(q, t)exists and is properly normalizable. Assume that the process qi(t, ω) is the solutionof differential equations

(3.1) qi(t, ω) = (dqi/dt)(t, ω) = vi(q(t, ω), t)

with random initial conditions qi(t0, ω) = qi0(ω). Once the joint distribution of

the random variables qi0(ω) is given the process qi(t, ω) is uniquely determined by

(3.1). One knows that in this situation ∂tρ + ∂i(ρvi) = 0 (continuity equation)with initial Cauchy data ρ(q, t) = ρ0(q). The natural origin of vi arises via a leastaction principle based on a Lagrangian L(q, q, t) with

(3.2) L∗(q, q, t) = L(q, q, t)− Φ(q, q, t); Φ =dS

dt= ∂tS + qi∂iS

Then vi(q, t) arises by minimizing

(3.3) I(t0, t1) = E[∫ t1

t0

L∗(q(t, ω), q(t, ω), t)dt]

where t0, t1 are arbitrary and E denotes the expectation (cf. [186, 187, 672,674, 698] for stochastic ideas). The minimum is to be achieved over the classof all random motions qi(t, ω) obeying (3.2) with arbitrarily varied velocity fieldvi(q, t) but having common initial values. One proves first

(3.4) ∂tS + H(q,∇S, t) = 0; vi(q, t) =∂H

∂pi(q,∇S(q, t), t)

Thus the value of I in (3.3) along the random curve qi(t, q0(ω)) is

(3.5) I(t1, t0, ω) =∫ t1

t0

L∗(q(, q0(ω)), q(t, q0(ω)), t)dt

Let µ(q0) denote the joint probability density of the random variables qi0(ω) and

then the expectation value of the random integral is

(3.6) I(t1, t0) = E[I(t1, t0, ω)] =∫Rn

∫ t1

t0

µ(q0)L∗(q(t, q0), q(t, q0), t)dnq0dt

Standard variational methods give then

(3.7) δI =∫Rn

dnq0µ(0)[∂L∗

∂qi(q(t1, q0), ∂tq(t1, q0), t)δqi(t1, q0)−

−∫ t1

t0

dt

(∂

∂t

∂L∗

∂qi(q(t, q0), ∂tq)t, q0), t)−

∂L∗

∂qi(q(t, q0), ∂tq(t, q0), t)

)δqi(t, q0)

]


where one uses the fact that µ(q0) is independent of time and δqi(t0, q0) = 0 (recallcommon initial data is assumed). Therefore

(3.8) (A) (∂L∗/∂qi)(q(t, q0), ∂tq(t, q0), t) = 0;

(B)∂

∂t

∂L∗

∂qi(q(t, q0), ∂tq(t, q0, t)−

∂L∗

∂qi(q(t, q0), ∂tq(t, q0), t) = 0

are the necessary conditions for obtaining a minimum of I. Conditions (B) are theusual Euler-Lagrange (EL) equations whereas (A) is a consequence of the fact thatin the most general case one must retain varied motions with δqi(t1, q0) differentfrom zero at the final time t1. Note that since L∗ differs from L by a total timederivative one can safely replace L∗ by L in (B) and putting (3.2) into (A) oneobtains the classical equations

(3.9) pi = (∂L/∂qi)(q(t, q0), q(t, q0), t) = ∂iS(q(t, q0), t)

It is known now that if det[(∂2L/∂qi∂qj ] = 0 then the second equation in (3.4) is aconsequence of the gradient condition (3.9) and of the definition of the Hamiltonianfunction H(q, p, t) = piq

i − L. Moreover (B) in (3.8) and (3.9) entrain the HJequation in (3.4). In order to show that the average action integral (3.6) actuallygives a minimum one needs δ2I > 0 but this is not necessary for Lagrangianswhose Hamiltonian H has the form

(3.10) HC(q, p, t) =1

2mgik(pi −Ai)(pk −Ak) + V

with arbitrary fields Ai and V (particle of mass m in an EM field A) which is theform for nonrelativistic applications; given positive definite gik such Hamiltoniansinvolve sufficiency conditions det[∂2L/∂qi∂qk] = mg > 0. Finally (B) in (3.8)with L∗ replaced by L) shows that along particle trajectories the EL equationsare satisfied, i.e. the particle undergoes a classical motion with probability one.Notice here that in (3.4) no explicit mention of generalized momenta is made;one is dealing with a random motion entirely based on position. Moreover theminimum principle (3.3) defines a 1-1 correspondence between solutions S(q, t)in (3.4) and minimizing random motions qi(t, ω). Provided vi is given via (3.4)the particle undergoes a classical motion with probability one. Thus once theLagrangian L or equivalently the Hamiltonian H is given, ∂tρ + ∂i(ρvi) = 0 and(3.4) uniquely determine the stochastic process qi(t, ω). Now suppose that somegeometric structure is given on M so that the notion of scalar curvature R(q, t) ofM is meaningful. Then we assume (ad hoc) that the actual Lagrangian is

(3.11) L(q, q, t) = LC(q, q, t) + γ(2/m)R(q, t)

where γ = (1/6)(n − 2)/(n − 1) with n = dim(M). Since both LC and R areindependent of we have L→ LC as → 0.

Now for a differential manifold with ds2 = gik(q)dqidqk it is standard that ina transplantation qi → qi + δqi one has δAi = Γi

kAdqk with Γi

k general affineconnection coefficients on M (Riemannian structure is not assumed). In [840]it is assumed that for = (gikAiAk)1/2 one has δ = φkdqk where the φk arecovariant components of an arbitrary vector (Weyl geometry). Then the actual


affine connections Γik can be found by comparing this with δ2 = δ(gikAiAk) and

using δAi = ΓikA

dqk. A little linear algebra gives then

(3.12) Γik = −

i

k

+ gim(gmkφ + gmφk − gkφm)

Thus we may prescribe the metric tensor gik and φi and determine via (3.12) theconnection coefficients. Note that Γi

k = Γik and for φi = 0 one has Riemannian

geometry. Covariant derivatives are defined via

(3.13) Ak,ı = ∂iA

k − ΓkA; Ak,i = ∂iAk + ΓkiA

for covariant and contravariant vectors respectively (where S,i = ∂iS). Note Ricci’slemma no longer holds (i.e. gik, = 0) so covariant differentiation and operationsof raising or lowering indices do not commute. The curvature tensor Ri

km in Weylgeometry is introduced via Ai

,k,−Ai,,k = F i

mkAm from which arises the standard

formula of Riemannian geometry

(3.14) Rimk = −∂Γi

mk + ∂kΓim + Γi

nΓnmk − Γi

nkΓnm

where (3.12) is used in place of the Christoffel symbols. The tensor Rimk obeys

the same symmetry relations as the curvature tensor of Riemann geometry as wellas the Bianchi identity. The Ricci symmetric tensor Rik and the scalar curvatureR are defined by the same formulas also, viz. Rik = R

ik and R = gikRik. Forcompleteness one derives here

(3.15) R = R + (n− 1)[(n− 2)φiφi − 2(1/

√g)∂i(

√gφi)]

where R is the Riemannian curvature built by the Christoffel symbols. Thus from(3.12) one obtains

(3.16) gkΓik = −gk

i

k

− (n− 2)φi; Γi

k = −

ik

+ nφk

Since the form of a scalar is independent of the coordinate system used one maycompute R in a geodesic system where the Christoffel symbols and all ∂gik vanish;then (3.12) reduces to Γi

k = φkκi + φδ

ik − gkφ

i and hence

(3.17) R = −gkm∂mΓik + ∂i(gkΓi

k) + gmΓinΓ

nmi − gmΓi

nΓnm

Further one has gmΓinΓ

nmi = −(n − 2)(φkφk) at the point in consideration.

Putting all this in (3.17) one arrives at

(3.18) R = R + (n− 1)(n− 2)(φkφk)− 2(n− 1)∂kφk

which becomes (3.15) in covariant form. Now the geometry is to be derived fromphysical principles so the φi cannot be arbitrary but must be obtained by the sameaveraged least action principle (3.3) giving the motion of the particle. The mini-mum in (3.3) is to be evaluated now with respect to the class of all Weyl geometrieshaving arbitrarily varied gauge vectors but fixed metric tensor. Note that once(3.11) is inserted in (3.2) the only term in (3.3) containing the gauge vector is thecurvature term. Then observing that γ > 0 when n ≥ 3 the minimum principle(3.3) may be reduced to the simpler form E[R(q(t, ω), t)] = min where only the


gauge vectors φi are varied. Using (3.15) this is easily done. First a little argu-ment shows that ρ(q, t) = ρ(q, t)/

√g transforms as a scalar in a coordinate change

and this will be called the scalar probability density of the random motion of theparticle (statistical determination of geometry). Starting from ∂tρ + ∂i(ρvi) = 0 amanifestly covariant equation for ρ is found to be ∂tρ+(1/

√g)∂i(

√gviρ) = 0. Now

return to the minimum problem E[R(q(t, ω), t)] = min; from (3.15) and ρ = ρ/√

gone obtains

(3.19) E[R(q(t, ω), t)] = E[R(q(t, ω), t)]+

+(n− 1)∫

M

[(n− 2)φiφi − 2(1/

√g)∂i(

√gφi)]ρ(q, t)

√gdnq

Assuming fields go to 0 rapidly enough on ∂M and integrating by parts one getsthen

(3.20) E[R] = E[R]− n− 1n− 2

E[gik∂i(log(ρ)∂k(log(ρ)]+

+n− 1n− 2

Egik[(n− 2)φi + ∂i(log(ρ)][(n− 2)φk + ∂k(log(ρ)]

Since the first two terms on the right are independent of the gauge vector and gik

is positive definite E[R] will be a minimum when

(3.21) φi(q, t) = −[1/(n− 2)]∂i[log(ρ)(q, t)]

This shows that the geometric properties of space are indeed affected by the pres-ence of the particle and in turn the alteration of geometry acts on the particlethrough the quantum force fi = γ(2/m)∂iR which according to (3.15) dependson the gauge vector and its derivatives. It is this peculiar feedback between thegeometry of space and the motion of the particle which produces quantum effects.

In this spirit one goes now to a geometrical derivation of the SE. Thus inserting(3.21) into (3.16) one gets

(3.22) R = R + (1/2γ√

ρ)[1/√

g)∂i(√

ggik∂k

√ρ)]

where the value (n−2)/6(n−1) for γ is used. On the other hand the HJ equation

(3.23) ∂tS + HC(q,∇S, t)− γ(2/m)R = 0

where (3.11) has been used. When (3.22) is introduced into (3.23) the HJ equationand the continuity equation ∂tρ + (1/

√g)(√

gviρ) = 0, with velocity field givenby (3.4), form a set of two nonlinear PDE which are coupled by the curvatureof space. Therefore self consistent random motions of the particle (i.e. randommotions compatible with (3.17)) are obtained by solving (3.23) and the continuityequation simultaneously. For every pair of solutions S(q, t, ρ(q, t)) one gets a pos-sible random motion for the particle whose invariant probability density is ρ. Thepresent approach is so different from traditional QM that a proof of equivalence

(3.4) can be written as


is needed and this is only done for Hamiltonians of the form (3.10) (which is notvery restrictive). The HJ equation corresponding to (3.10) is

(3.24) ∂tS +1

2mgik(∂iS −Ai)(∂kS −Ak) + V − γ

2

mR = 0

with R given by (3.22). Moreover using (3.4) as well as (3.10) the continuityequation becomes

(3.25) ∂tρ + (1/m√

g)∂i[ρ√

ggik(∂kS −Ak)] = 0

Owing to (3.22),(3.24) and (3.25) form a set of two nonlinear PDE which mustbe solved for the unknown functions S and ρ. Now a straightforward calculationsshows that, setting

(3.26) ψ(q, t) =√

ρ(q, t)exp](i/)S(q, t)],

the quantity ψ obeys a linear PDE (corrected from [840])

(3.27) i∂tψ =1

2m

[i∂i

√g

√g

+ Ai

]gik(i∂k + Ak)

ψ +

[V − γ

2

mR

]ψ

where only the Riemannian curvature R is present (any explicit reference to thegauge vector φi having disappeared). (3.27) is of course the SE in curvilinearcoordinates whose invariance under point transformations is well known. Moreover(3.26) shows that |ψ|2 = ρ(q, t) is the invariant probability density of finding theparticle in the volume element dnq at time t. Then following Nelson’s argumentsthat the SE together with the density formula contains QM the present theory isphysically equivalent to traditional nonrelativistic QM. One sees also from (3.26)and (3.27) that the time independent SE is obtained via S = S0(q) − Et withconstant E and ρ(q). In this case the scalar curvature of space becomes timeindependent; since starting data at t0 is meaningless one replaces the continuityequation with a condition

∫M

ρ(q)√

gdnq = 1.

REMARK 3.3.1. We recall (cf. [188]) that in the nonrelativistic contextthe quantum potential has the form Q = −(2/2m)(∂2√ρ/

√ρ) (ρ ∼ ρ here) and

in more dimensions this corresponds to Q = −(2/2m)(∆√

ρ/√

ρ). Here we havea SE involving ψ =

√ρexp[(i/)S] with corresponding HJ equation (3.24) which

corresponds to the flat space 1-D St + (s′)2/2m + V + Q = 0 with continuityequation ∂tρ + ∂(ρS′/m) = 0 (take Ak = 0 here). The continuity equation in(3.25) corresponds to ∂tρ + (1/m

√g)∂i[ρ

√ggik(∂kS)] = 0. For Ak = 0 (3.24)

becomes

(3.28) ∂tS + (1/2m)gik∂iS∂kS + V − γ(2/m)R = 0

This leads to an identification Q ∼ −γ(2/m)R where R is the Ricci scalar in theWeyl geometry (related to the Riemannian curvature built on standard Christoffelsymbols via (3.15)). Here γ = (1/6)[(n − 2)/(n − 2)] as above which for n = 3becomes γ = 1/12; further the Weyl field φi = −∂ilog(ρ). Consequently (seebelow).

PROPOSITION 3.1. For the SE (3.27) in Weyl space the quantum potentialis Q = −(2/12m)R where R is the Weyl-Ricci scalar curvature. For Riemannianflat space R = 0 this becomes via (3.22)

(3.29) R =1

2γ√

ρ∂ig

ik∂k√

ρ ∼ 12γ

∆√

ρ√

ρ⇒ Q = − 2

2m

∆√

ρ√

ρ

as it should and the SE (3.27) reduces to the standard SE in the form i∂tψ =−(2/2m)∆ψ + V ψ (Ak = 0).

3.1. FISHER INFORMATION REVISITED. Via Remarks 1.1.4, 1.1.5,and 1.1.6 from Chapter 1 (based on [395, 446, 447, 448, 449, 805, 806]) werecall the derivation of the SE in Theorem 1.1.1. Thus with some repetition recallfirst that the classical Fisher information associated with translations of a 1-Dobservable X with probability density P (x) is

(3.30) FX =∫

dx P (x)([log(P (x)]′)2 > 0

One has a well known Cramer-Rao inequality V ar(X) ≥ F−1X where V ar(X) ∼

variance of X. A Fisher length for X is defined via δX = F−1/2X and this quantifies

the length scale over which p(x) (or better log(p(x))) varies appreciably. Then theroot mean square deviation ∆X satisfies ∆X ≥ δX. Let now P be the momentumobservable conjugate to X, and Pcl a classical momentum observable correspondingto the state ψ given via pcl(x) = (/2i)[(ψ′/ψ)−(ψ′/ψ)]. One has then the identityψ=< pcl >ψ following via integration by parts. Now define the nonclassicalmomentum by pnc = p− pcl and one shows then

(3.31) ∆X∆p ≥ δX∆p ≥ δX∆pnc = /2

Then consider a classical ensemble of n-dimensional particles of mass m movingunder a potential V. The motion can be described via the HJ and continuityequations

(3.32)∂s

∂t+

12m|∇s|2 + V = 0;

∂P

∂t+∇ ·

[P∇s

m

]= 0

for the momentum potential s and the position probability density P (note thatthere is no quantum potential and this will be supplied by the information term).These equations follow from the variational principle δL = 0 with LagrangianL =

∫dt dnx P

[(∂s/∂t) + (1/2m)|∇s|2 + V

]. It is now assumed that the classical

Lagrangian must be modified due to the existence of random momentum fluctua-tions. The nature of such fluctuations is immaterial and one can assume that themomentum associated with position x is given by p = ∇s+N where the fluctuationterm N vanishes on average at each point x. Thus s changes to being an averagemomentum potential. It follows that the average kinetic energy < |∇s|2 > /2mappearing in the Lagrangian above should be replaced by < |∇s + N |2 > /2mgiving rise to

(3.33) L′ = L + (2m)−1

∫dt < N ·N >= L + (2m)−1

∫dt(∆N)2

where ∆N =< N ·N >1/2 is a measure of the strength of the quantum fluctuations.The additional term is specified uniquely, up to a multiplicative constant, by thethree assumptions given in Remark 1.1.4 This leads to the result that

(3.34) (∆N)2 = c

∫dnx P |∇log(P )|2

where c is a positive universal constant (cf. [446]). Further for = 2√

c andψ =

√Pexp(is/) the equations of motion for p and s arising from δL′ = 0 are

i∂ψ∂t = − 2

2m∇2ψ + V ψ.

A second derivation is given in Remark 1.1.5. Thus let P (yi) be a probabilitydensity and P (yi + ∆yi) be the density resulting from a small change in the yi.Calculate the cross entropy via

(3.35) J(P (yi + ∆yi) : P (yi)) =∫

P (yi + ∆yi)logP (yi + ∆yi)

P (yi)dny

[12

∫1

P (yi)∂P (yi)

∂yi

∂P (yi)∂yk)

dny

]∆yi∆yk = Ijk∆yi∆yk

The Ijk are the elements of the Fisher information matrix. The most generalexpression has the form

(3.36) Ijk(θi) =12

∫1

P (xi|θi)∂P (xi|θi)

∂θj

∂P (xi|θi)∂θk

dnx

where P (xi|θi) is a probability distribution depending on parameters θi in additionto the xi. For P (xi|θi) = P (xi + θi) one recovers (3.35). If P is defined overan n-dimensional manifold with positive inverse metric gik one obtains a naturaldefinition of the information associated with P via


2

∫1P

∂P

∂yi

∂P

∂ykdny

Now in the HJ formulation of classical mechanics the equation of motion takes theform

(3.38)∂S

∂t+

12gµν ∂S

∂xµ

∂S

∂xν+ V = 0

where gµν = diag(1/m, · · · , 1/m). The velocity field uµ is then uµ = gµν(∂S/∂xν).When the exact coordinates are unknown one can describe the system by meansof a probability density P (t, xµ) with

∫Pdnx = 1 and

(3.39) (∂P/∂t) + (∂/∂xµ)(Pgµν(∂S/∂xν) = 0

These equations completely describe the motion and can be derived from theLagrangian

(3.40) LCL =∫

P (∂S/∂t) + (1/2)gµν(∂S/∂xµ)(∂S/∂xν) + V dtdnx


using fixed endpoint variation in S and P. Quantization is obtained by adding aterm proportional to the information I defined in (3.37). This leads to(3.41)


P

∂S

∂t+

12gµν

[∂S

∂xµ

∂S

∂xν+

λ

P 2

∂P

∂xµ

∂P

∂xν

]+ V

dtdnx

Fixed endpoint variation in S leads again to (3.39) while variation in P leads to

(3.42)∂S

∂t+

12gµν

[∂S

∂xµ

∂S

∂xν+ λ

(1

P 2

∂P

∂xµ

∂P

∂xν− 2

P

∂2P

∂xµ∂xν

)]+ V = 0

These equations are equivalent to the SE if ψ =√

Pexp(iS/) with λ = (2)2

(recall also Remark 1.1.6 for connections to entropy). Now following ideas in[219, 223, 715] we note in (3.41) for φµ ∼ Aµ = ∂µlog(P ) (which arises in (3.21))and pµ = ∂uS, a complex velocity can be envisioned leading to (cf. also [224])

(3.43) |pµ + i√

λAµ|2 = p2µ + λA2

µ ∼ gµν

(∂S

∂xµ

∂S

∂xν+

λ

P 2

∂P

∂xµ

∂P

∂xν

)Further I in (3.37) is exactly known from φµ so one has a direct connection betweenFisher information and the Weyl field φµ, along with motivation for a complexvelocity (cf. Sections 1.2 and 1.3).

REMARK 3.3.2. Comparing now with [189] and quantum geometry in theform ds2 =

∑(dp2

j/pj) on a space of probability distributions (to be discussedin Chapter 5) we can define (3.37) as a Fisher information metric in the presentcontext. This should be positive definite in view of its relation to (∆N)2 in (3.34)for example. Now for ψ = Rexp(iS/) one has (ρ ∼ ρ here)

(3.44) − 2

2m

R′′

R≡ − 2

2m

∂2√ρ√

ρ= − 2

8m

[2ρ′′

ρ−(

ρ′

ρ

)2]

in 1-D while in more dimensions we have a form (ρ ∼ P )

(3.45) Q ∼ −22gµν

[1

P 2

∂P

∂xµ

∂P

∂xν− 2

P

∂2P

∂xµ∂xν

]as in (3.44) (arising from the Fisher metric I of (3.37) upon variation in P in theLagrangian). It can also be related to an osmotic velocity field u = D∇log(ρ)via Q = (1/2)u2 + D∇ · u connected to Brownian motion where D is a diffusioncoefficient (cf. [223, 395, 715]). For φµ = −∂µlog(P ) we have then u = −Dφwith Q = D2((1/2)(| |2−∇·φ), expressing Q directly in terms of the Weyl vector.This enforces the idea that QM is built into Weyl geometry!

3.2. THE KG EQUATION. The formulation above from [840] was mod-ified in [841] to a derivation of the Klein-Gordon (KG) equation via an averageaction principle. The spacetime geometry was then obtained from the average ac-tion principle to obtain Weyl connections with a gauge field φµ (thus the geometryhas a statistical origin). The Riemann scalar curvature R is then related to theWeyl scalar curvature R via an equation

(3.46) R = R− 3[(1/2)gµνφµφν + (1/√−g)∂µ(

√−ggµνφν)]

φ


Explicit reference to the underlying Weyl structure disappears in the resulting SE(as in (3.27)). The HJ equation in [841] has this form (for Aµ = 0 and V = 0)gµν∂µS∂νS = m2 − (R/6) so in some sense (recall here = c = 1) m2 − (R/6) ∼M2 where M2 = m2exp(Q) and Q = (2/m2c2)(

√ρ/√

ρ) ∼ (√

ρ/m2√ρ) viaSection 3.2 (for signature (−,+,+,+) - recall here gµν∇µS∇νS = M2c2). Thusfor exp(Q) ∼ 1 + Q one has m2 − (R/6) ∼ m2(1 + Q) ⇒ (R/6) ∼ −Qm2 ∼−(

√ρ/√

ρ). This agrees also with [219] where the whole matter is analyzedincisively (cf. also Remark 3.3.5). We recall also here from [798] (cf. Section3.2.2) that in the conformal geometry the particles do not follow geodesics of theconformal metric alone. We will sketch an elaboration of this now from [841](paper one). Thus summarizing [840] and the second paper in [841] one showsthat traditional QM is equivalent (in some sense) to classical statistical mechanicsin Weyl spaces. The following two points of view are taken to be equivalent

(1) (A) The spacetime is a Riemannian manifold and the statistical behaviorof a spinless particle is described by the KG equation while probabilitiescombine according to Feynman quantum rules.

(2) (B) The spacetime is a generic affinely connected manifold whose actualgeometric structure is determined by the matter content. The statis-tical behavior of a spinless particle is described by classical statisticalmechanics and probabilities combine according to Laplace rules.

(3) In nonrelativistic applications the words spacetime, Riemannian, and KGare to be replacedby space, Euclidean, and SE.

We are skipping over the second paper in [841] here and going to the first pa-per which treates matters in a gauge invariant manner. The moral seems to be(loosly) that quantum mechanics in Riemannian spacetime is the same as clas-sical statistical mechanics in a Weyl space. In particular one wants to establishthat traditional QM, based on wave equations and ad hoc probability calculus(as in (1) above) is merely a convenient mathematical construction to overcomethe complications arising from a nontrivial spacetime geometric structure. Hereone works from first principles and includes gauge invariance (i.e. invariance withrespect to an arbitrary choice of the spacetime calibration). The spacetime issupposed to be a generic 4-dimensional differential manifold with torsion free con-nections Γλ

µν = Γλνµ and a metric tensor gµν with signature (+,−,−,−) (one takes

= c = 1 - which I deem unfortunate since the role and effect of such quantitiesis not revealed). Here the (restrictive) hypothesis of assuming a Weyl geometryfrom the beginning is released, both the particle motion and the spacetime geo-metric structure are derived from a single average action principle. A result ofthis approach is that the spacetime connections are forced to be integrable Weylconnections by the extremization principle.

The particle is supposed to undergo a motion in spacetime with determin-istic trajectories and random initial conditions taken on an arbitrary spacelike3-dimensional hypersurface; thus the theory describes a relativistic Gibbs ensem-ble of particles (cf. Remark 3.3.3). Both the particle motion and the spacetime


connections can be obtained from the average stationary action principle

(3.47) δ

[E

(∫ τ2

τ1

L(x(τ, x(τ))dτ

)]= 0

This action integral must be parameter invariant, coordinate invariant, and gaugeinvariant. All of these requirements are met if L is positively homogeneous of thefirst degree in xµ = dxµ/dτ and transforms as a scalar of Weyl type w(L) = 0.The underlying probability measure must also be gauge invariant. A suitableLagrangian is then

(3.48) L(x, dx) = (m2 − (R/6))1/2ds + Aµdxµ

where ds = (gµν xµxν)1/2dτ is the arc length and R is the space time scalar cur-vature; m is a parameterlike scalar field of Weyl type (or weight) w(m) = −(1/2).The factor 6 is essentially arbitrary and has been chosen for future convenience.The vector field Aµ can be interpreted as a 4-potential due to an externally appliedEM field and the curvature dependent factor in front of ds is an effective particlemass. This seems a bit ad hoc but some feeling for the nature of the Lagrangiancan be obtained from Section 3.2 (cf. also [63]). The Lagrangian will be gaugeinvariant provided the Aµ have Weyl type w(Aµ) = 0. Now one can split Aµ intoits gradient and divergence free parts Aµ = Aµ−∂µS, with both S and Aµ havingWeyl type zero, and with Aµ interpreted as and EM 4-potential in the Lorentzgauge. Due to the nature of the action principle regarding fixed endpoints in vari-ation one notes that the average action principle is not invariant under EM gaugetransformations Aµ → Aµ + ∂µS; but one knows that QM is also not invariantunder EM gauge transformations (cf. [17]) so there is no incompatability withQM here.

Now the set of all spacetime trajectories accessible to the particle (the particlepath space) may be obtained from (3.47) by performing the variation with respectto the particle trajectory with fixed metric tensor, connections, and an underlyingprobability measure. Thus (cf. Remark 3.3.3) the solution is given by the so-calledCaratheodory complete figure (cf. [826]) associated with the Lagrangian

(3.49) L(x, dx) = (m2 − (R/6))1/2ds + Aµdxµ

(note this leads to the same equations as (3.48) since the Lagrangians differ by atotal differential dS). The resulting complete figure is a geometric entity formedby a one parameter family of hypersurfaces S(x) = const. where S satisfies the HJequation

(3.50) gµν(∂µS − Aµ)(∂νS − Aν) = m2 − R

6and by a congruence of curves intersecting this family given by

(3.51)dxµ

ds=

gµν(∂νS − Aν)[gρσ(∂ρS − Aρ)(∂σS − Aσ)]1/2

The congruence yields the actual particle path space and the underlying proba-bility measure on the path space may be defined on an arbitrary 3-dimensionalhypersurface intersecting all of the members of the congruence without tangencies


(cf. [443]). The measure will be completely identified by its probability currentdensity jµ (see [841] and Remark 3.3.3). Moreover, since the measure is inde-pendent of the arbitrary choice of the hypersurface, jµ must be conserved, i.e.∂µjµ = 0 (see Remark 3.3.3). Since the trajectories are deterministically definedby (3.51), jµ must be parallel to the particle 4-velocity (3.51), and hence

(3.52) jµ = ρ√−ggµν(∂νS − Aν)

with some ρ > 0. Now gauge invariance of the underlying measure as well as ofthe complete figure requires that jµ transforms as a vector density of Weyl typew(jµ) = 0 and S as a scalar of Weyl type w(S) = 0. From (3.52) one sees thenthat ρ transforms as a scalar of Weyl type w(ρ) = −1 and ρ is called the scalarprobability density of the particle random motion.

The actual spacetime affine connections are obtained from (3.47) by perform-ing the variation with respect to the fields Γλ

µν for a fixed metric tensor, particletrajectory, and probability measure. It is expedient to tranform the average actionprinciple to the form of a 4-volume integral

(3.53) δ

[∫Ω

d4x[(m2 − (R/6))(gµνjµjν ]1/2 + Aµjµ

]= 0

where Ω is the spacetime region occupied by the congruence (3.51) and jµ is givenby (3.52) (cf. [841] and Remark 3.3.3 for proofs). Since the connection fields Γλ

µν

are contained only in the curvature term R the variational problem (3.53) can befurther reduced to

(3.54) δ

[∫Ω

ρR√−gd4x

]= 0

(here the HJ equation (3.50) has been used). This states that the average space-time curvature must be stationary under a variation of the fields Γλ

µν (principle ofstationary average curvature). The extremal connections Γλ

µν arising from (3.54)are derived in [841] using standard field theory techniques and the result is

(3.55) Γλµν =

λ

µ ν

+

12(φµδλ

ν + φνδλµ − gµνgλρφρ); φµ = ∂µlog(ρ)

This shows that the resulting connections are integrable Weyl connections with agauge field φµ (cf. [840], Section 3, and Section 3.1). The HJ equation (3.50) andthe continuity equation ∂µjµ = 0 can be consolidated in a single complex equationfor S, namely

(3.56) eiSgµν(iDµ − Aµ)(iDν − Aν)e−iS − (m2 − (R/6)) = 0; Dµρ = 0

Here Dµ is (doubly covariant - i.e. gauge and coordinate invariant) Weyl derivativegiven by (cf. [63])

(3.57) DµTαβ = ∂µTα

β + ΓαµεT

εβ − Γε

µβTαε + w(T )φµTα

β

It is to be noted that the probability density (but not the rest mass) remainsconstant relative to Dµ. When written out (3.56) for a set of two coupled partialdifferential equations for ρ and S. To any solution corresponds a particular randommotion of the particle.


Next one notes that (3.56) can be cast in the familiar KG form, i.e.

(3.58) [(i/√−g)∂µ

√−g − Aµ]gµν(i∂ν − Aν)ψ − (m2 − (R/6))ψ = 0

where ψ =√

ρexp(−iS) and R is the Riemannian scalar curvature built out of gµν

only. We have the (by now) familiar formula

(3.59) R = R− 3[(1/2)gµνφµφν + (1/√−g)∂µ(

√−ggµνφν)]

According to point of view (A) above in the KG equation (3.58) any explicitreference to the underlying spacetime Weyl structure has disappeared; thus theWeyl structure is hidden in the KG theory. However we note that no physicalmeaning is attributed to ψ or to the KG equation. Rather the dynamical andstatistical behavior of the particle, regarded as a classical particle, is determinedby (3.56), which, although completely equivalent to the KG equation, is expressedin terms of quantities having a more direct physical interpretation.

REMARK 3.3.3. We extract here from the Appendices to paper 1 of [841].In Appendix A one shows that the Caratheodory complete figure formed by thecongruence (3.51) solves the variational problem (3.47). One needs the notion ofthe Gibbs ensemble in relativistic mechanics (cf. [443]). Roughly a relativisticGibbs ensemble of particles may be assimilated to an incoherent globule of mattermoving in spacetime. More precisely a relativistic Gibbs ensemble is given by(i) A congruence of timelike curves in spacetime (the path space of the particles)and (ii) A probability measure defined on this congruence (note a congruence ofspacelike curves could also be envisioned but causality is affected - a physicalintepretation is unclear although it could be related to a statistical formulation ofvirtual phenomena). The construction here goes as follows. Let K be a 3-parametercongruence of time like curves in spacetime be given via () xµ = xµ(τ, uk) wherek = 1, 2, 3 and τ is an arbitrary parameter along each curve of the congruence.For simplicity assume that the congruence covers a region Ω of spacetime simply(i.e. one and only one curve of K passes through each point of Ω). Then onecan regard () as a change of coordinates from xµ to yµ where y0 = t, yk = uk

(assume the Jacobian is nonzero in Ω). Consider then the action integral L =∫ τ2

τ1L(x(τ, uk), x(τ, uk)dτ with L homogeneous of the first degree in the derivatives

xµ = ∂xµ/∂τ . Given a 1-1 correspondence between the uk and members of thecongruence K one may introduce a formula for the probability that the particlefollows a sample path having parameters uk in some 3-dimensional region B asprob(B) =

∫B⊂R

µ(uk)du1du2du3 where µ(uk) is some probability density definedon R3. Hence the average action integral in (3.47) may be written as

(3.60) I = E

[∫ τ2

τ1

Ldτ

]=∫R3

∫ τ2

τ1

µ(uk)L(xµ(τ, uk), xµ(τ, uk)dτ∏

dui

The last term is a 4-dimensional volume integral over the zone between the hyper-planes y0 = τ1 and y0 = τ2 in the y coordinate. In the x coordinates these hyper-planes are mapped on two 3-dimensional hypersurfaces τ(xµ) = τ1 and τ(xµ) = τ2

where τ(xµ) is obtained by solving () with respect to τ ; since they are merely aresult of the parametrization of K they can be regarded as essentially arbitrary.


The integrand in (3.60) depends on the 4 unknown functions xµ(yν) and on theirfirst derivatives ∂xµ/∂y0, and on the coordinates yν themselves. Therefore thevariational problem δI = 0 is reduced to a standard variational problem whosesolution will yield the functions xµ(τ, uk), i.e. the actual congruence that rendersthe average action stationary.

Now the Lagrangian density in (3.60) is Λ = µ(uk)L(xµ(τ, uk), xµ,τ (τ, uk) in

which xµ,τ = xµ with τ and uk are the independent variables. By standard methods

the EL expressions are (xµ,k = ∂xµ/∂uk)

(3.61) E(Λ) =∂

∂uk

[∂Λ∂xµ

,k

]+

∂

∂τ

[∂Λ∂xµ

,τ

]− ∂Λ

∂xµ

In this case howver ∂Λ/∂xµ,k = 0 and hence the fixed equations E(Λ) = 0 reduce

to (note µ does not depend explicitly on τ)

(3.62)∂

∂τ

[µ

∂L

∂xµ,τ

]− µ

∂L

∂xµ= 0⇒ ∂

∂τ

[∂L

∂xµ

]− ∂L

∂xµ= 0

and this coincides with the EL equations associated with the action integral above.This means that the actual congruence must be a congruence of extremals orequivalently that the particle obeys equations of motion (3.62) with probabilityone. Even if the congruence is extremal however we are left with nonvanishingsurface terms in the variation of I, namely(3.63)

δI =∫R3

µ(uk)∏

dui

[∂L

∂xµ(τ2, u

k)δxµ(τ2, uk)− ∂L

∂xµ(τ1, u

k)δxµ(τ1, uk)]

= 0

In (3.63) the quantities δxµ at τ = τ2 and τ = τ1 are displacements between pointsP and P + δP where the curves xµ and xµ + δxµ intersect the hypesurfaces τ = τ2

and τ = τ1 so δxµ(τ1, uk) and δxµ(τ1, u

k) are tangential to the hypersurfaces.Since the hypersurfaces τ(xµ) = const. are essentially arbitrary so must be thedisplacements δxµ and δI = 0 implies then (•) ∂L/∂xµ(τ, uk) = 0. Finally relatingL with the Lagrangian (3.48) and comparing with L as defined in (3.49) one has∂L/∂xµ = ∂L/∂xµ − ∂µS so (•) yields ∂L/∂xµ = ∂µS. Moreover L and L,differing only by a total differential dS, lead to the same EL equations and henceone can replace L by L in (3.62). In conclusion the congruence that renders theaverage action stationary must be (i) A congruence of curves that are extremalwith respect to Lagrangian L and (ii) A congruence satisfying the integrabilityconditions ∂L/∂Xµ = ∂µS. However by standard HJ theory such a congruence isgiven by (3.51) provided S(xµ) obeys the HJ equation associated with L, namely(3.50).

In appendix B the current density jµ is introduced and the equivalence betweenthe average action (3.47) and the 4-volume integral (3.53) is proved. This providesa useful connection between ensemble averages and 4-volume integrals appearingin field theories. Here (3.60) is expressed in terms of the y coordinates (τ, uk) andit can also be expressed in terms of the x coordinates. For this one introduces the


current density jµ associated with the relativistic Gibbs ensemble. The surfaceelement normal to the hypersurface τ(uk) = const. is given by dσµ = πµdu1du2du3

where πµ are Jacobians

(3.64) π0 =∂(x1, x2, x3)∂(u1, u2, u3)

; π1 =∂(x0, x2, x3)∂(u1, u2, u3)

, · · ·

Then define the current density via µ = jµπµ so that prob(B) becomes

(3.65) prob(B) =∫

B⊂R3µdu1du2du3 =

∫B⊂R3

jµdσµ

The direction of jµ is still not defined so one is free to choose the current directionparallel to the congruence K, i.e. jµ = λxµ. The independence of the underlyingmeasure on the chosen hypersurface τ = const. is exprssed analytically by the factthat µ = µ(u1, u2, u3) does not depend on τ explicitly. Consequently ∂µjµ = 0since by the Gauss theorem

(3.66)∫

τ(xµ)=τ2

jµdσµ −∫

τ(xµ)=τ1

jµdσµ =∫

Ω

∂µjµd4x = 0

where Ω is the strip between the essentially arbitrary hypersurfaces τ = τ1 andτ = τ2. The same result could be obtained by differentiating µ = jµπµ and usingproperties of Jacobians. Passing to x coordinates (3.60) becomes

(3.67) I =∫

Ω

µLJ−1d4x; J =∂(x0, x1, x2, x3)∂(τ, u1, u2, u3)

Note that by definition J = (∂xµ/∂τ)πµ so

(3.68) I =∫

Ω

µ[L(xµ, xµ)/(xµπµ)]d4x

Since L is homogeneous of the first degree in the xµ the term in square brackets in(3.68) is homogeneous of degree zero in the xµ. Hence we can replace xµ with thecurrent jµ = λxµ without affecting the integral to obtain I =

∫Ω

L(xµjµ)d4x whereµ = jµπµ has been used. Thus the average action I may be converted to a fourvolume integral of L(xµ, jµ). When this formal substitution is made in (3.48),(3.53) is obtained. This substitution does not alter the functional dependenceof the average action integral I on the connection fields Γλ

µν so the variationalproblems (3.47) and (3.53) are equivalent as long as the variation is performedwith respect to these fields.

In Appendix C one derives (3.55); since similar calculations have already beenused earlier (and will recur again) we omit this here.

REMARK 3.3.4. The formula (3.59) goes back to Weyl [986] and theconnection of matter to geometry arises from (3.55). The time variable is treatedin a special manner here related to a Gibbs ensemble and ρ > 0 is built intothe theory. Thus problems of statistical transparancy as in Remark 2.3.3 willapparently not arise.

REMARK 3.3.5. As mentioned at the beginning of Section 3.2, in [219]


the Santamato theory is analyzed in depth from several points of view and anumber of directions for further study are indicated (in [224] the importance ofa complex velocity is emphasized - see also Section 7.1.2). There is also a relateddevelopment for the Dirac equation using an approach related to [397, 463, 464],where both relativistic and nonrelativistic spin 1/2 particles can be classicallytreated using anticommuting Grassmanian variables. However we prefer to treatthe Dirac equation in a different manner later (cf. also [89] and Section 2.1.1).

4. SCALE RELATIVITY AND KG

In [186] and Section 1.2 we sketched a few developments in the theory ofscale relativity. This is by no means the whole story and we want to give a tasteof some further main ideas while deriving the KG equation in this context (cf.[11, 232, 272, 273, 274, 715, 716, 717, 718, 719, 720, 721]). A main ideahere is that the Schrodinger, Klein-Gordon, and Dirac equations are all geodesicequations in the fractal framework. They have the form D2/ds2 = 0 where D/dsrepresents the appropriate covariant derivative. The complex nature of the SE andKG equation arises from a discrete time symmetry breaking based on nondifferen-tiability. For the Dirac equation further discrete symmetry breakings are neededon the spacetime variables in a biquaternionic context (cf. here [232]). First we goback to [715, 716, 720] and sketch some of the fundamentals of scale relativity.This is a very rich and beautiful theory extending in both spirit and generalitythe relativity theory of Einstein (cf. also [225] for variations involving Cliffordtheory). The basic idea here is that (following Einstein) the laws of nature applywhatever the state of the system and hence the relevant variables can only bedefined relative to other states. Standard scale laws of power-law type correspondto Galilean scale laws and from them one actually recovers quantum mechanics(QM) in a nondifferentiable space. The quantum behavior is a manifestation ofthe fractal geometry of spacetime. In particular (as indicated in Section 1.2) thequantum potential is a manifestation of fractality in the same way as the Newtonpotential is a manifestation of spacetime curvature. In this spirit one can alsoconjecture (cf. [720]) that this quantum potential may explain various dynamicaleffects presently attributed to dark matter (cf. also [16] and Chapter 4). Now forbasics one deals with a continuous but nondifferentiable physics. It is known forexample that the length of a continuous nondifferentiable curve is dependent onthe resolution ε. One approach now involves smoothing a nondifferentiable func-tion f via f(x, ε) =

∫∞−∞ φ(x, y, ε)f(y)dy where φ is smooth and say “centered”

at x (we refer also to Remark 1.2.8 and [11, 272, 273, 274] for a more refinedtreatment of such matters). There will now arise differential equations involving∂f/∂log(ε) and ∂2f/∂x∂log(ε) for example and the log(ε) term arises as follows.Consider an infinitesimal dilatation ε → ε′ = ε(1 + dρ) with a curve length

(4.1) (ε) → (ε′) = (ε + εdρ) = (ε) + εεdρ = (1 + Ddρ)(ε)

Then D = ε∂ε = ∂/∂log(ε) is a dilatation operator and in the spirit of renormaliza-tion (multiscale approach) one can assume ∂(x, ε)/∂log(ε) = β() (where (x, ε)refers to the curve defined by f(x, ε)). Now for Galilean scale relativity consider

4. SCALE RELATIVITY AND KG 127

∂(x, ε)/∂log(ε) = a + b which has a solution

(4.2) (x, ε) = 0(x)

[1 + ζ(x)

(λ

ε

)−b]

where λ−bζ(x) is an integration constant and 0 = −a/b. One can choose ζ(x) sothat < ζ2(x) >= 1 and for a = 0 there are two regimes (for b < 0)

(1) ε << λ ⇒ ζ(x)(λ/ε)−b >> 1 and is given by a scale invariant fractallike power with dimension D = 1− b, namely (x, ε) = 0(λ/ε)−b.

(2) ε >> λ⇒ ζ(x)(λ/ε)−b << 1 and is independent of scale.

Here ε = λ constitutes a transition point between fractal and nonfractal behavior.Only the special case a = 0 yields unbroken scale invariance of = 0(λ/ε)δ (δ =−b) and one has then D = b so the scale dimension is an eigenvalue of D. Finallythe case b > 0 corresponds to the cosmological domain.

Now one looks for scale covariant laws and checks this for power laws φ =φ0(λ/ε)δ. Thus a scale transformation for δ(ε′) = δ(ε) will have the form

(4.3) logφ(ε′)φ0

= logφ(ε)φ0

+ V δ(ε); V = logε

ε′

In the same way that only velocity differences have a physical meaning in Galileanrelativity here only V differences or scale differences have a physical meaning.Thus V is a “state of scale” just as velocity is a state of motion. In this spiritlaws of linear transformation of fields in a scale transformation ε → ε′ amount tofinding A,B,C,D(V ) such that

(4.4) logφ(ε′)φ0

= A(V )logφ(ε)φ0

+ B(V )δ(ε); δ(ε′) = C(V )logφ(ε)φ0

+ D(V )δ(ε)

Here A = 1, B = V, C = 0, D = 1 corresponds to the Galileo group. Note alsoε → ε′ → ε′′ ⇒ V ′′ = V + V ′. Now for the analogue of Lorentz transforma-tions there is a need to preserve the Galilean dilatation law for scales larger thanthe quantum classical transition. Note V = log(ε/ε′) ∼ ε/ε′ = exp(−V ) and setρ = ε′/ε with ρ′ = ε′′/ε′ and ρ′′ = ε′′/ε; then logρ′′ = log ρ + log ρ′ and one isthinking here of ρ : ε → ε′, ρ′ : ε′ → ε′′ and ρ′′ : ε → ε′′ with compositions(the notation is meant to somehow correspond to (4.1)). Now recall the Einstein-Lorentz law w = (u+v)/[1+(uv/c2)] but one now has several regimes to consider.Following [716, 720] small scale symmetry is broken by mass via the emergenceof λc = /mc (Compton length) and λdB = /mν (deBroglie length), while forextended objects λth = /m < ν2 >1/2 (thermal deBroglie length) affects transi-tions. The transition scale in (4.2) is the Einstein-deBroglioe scale (in rest frameλ ∼ τ = /mc2) and in the cosmological realm the scale symmetry is broken bythe emergence of static structure of typical size λg = (1/3)(GM/ < ν2 >). Thescale space consists of three domains (quantum, classical - scale independent, andcosmological). Another small scale transition factor appears in the Planck lengthscale λP = (G/c3)1/2 and at large scales the cosmological constant Λ comes into

play. With this background the composition of dilatations is taken to be

(4.5) logε′

λ=

logρ + log ελ

1 + logρlog ελ

log2(L/λ)

=logρ + log ε

λ

1 + logρlog(ε/λ)C2

where L ∼ λP near small scales and L ∼ Λ near large scales (note ε = L⇒ ε′ = Lin (A.4)). Comparing with w = (u + v)/(1 + (uv/c2)) one thinks of log(L/λ) =C ∼ c (note here log2(a/b) = log2(b/a) in comparing formulas in [716, 720]).Lengths now change via

(4.6) log′

0=

log(/0) + δlogρ√1− log2 ρ

C2

and the scale variable δ (or djinn) is no longer constant but changes via

(4.7) δ(ε′) =δ(ε) + logρlog(/0)

C2√1− log2ρ

C2

where λ ∼ fractal-nonfractal transition scale.

We have derived the SE in Section 1.2 (cf. also [186]) and go now to the KGequation via scale relativity. The derivation in the first paper of [232] seems themost concise and we follow that at first (cf. also [716]). All of the elements ofthe approach for the SE remain valid in the motion relativistic case with the timereplaced by the proper time s, as the curvilinear parameter along the geodesics.Consider a small increment dXµ of a nondifferentiable four coordinate along oneof the geodesics of the fractal spacetime. One can decompose this in terms ofa large scale part LS < dXµ >= dxµ = vµds and a fluctuation dξµ such thatLS < dξµ >= 0. One is led to write the displacement along a geodesic of fractaldimension D = 2 via

(4.8) dXµ± = d±xµ + dξµ

± = vµ±ds + uµ

±√

2Dds1/2

Here uµ± is a dimensionless fluctuation andd the length scale 2D is introduced for

dimensional purposes. The large scale forward and backward derivatives d/ds+

and d/ds− are defined via

(4.9)d

ds±f(s) = lims→0±LS

⟨f(s + δs)− f(s)

δs

⟩Applied to xµ one obtains the forward and backward large scale four velocities ofthe form

(4.10) (d/dx+)xµ(s) = vµ+; (d/ds−)xµ = vµ

−

Combining yields

(4.11)d′

ds=

12

(d

ds++

d

ds−

)− i

2

(d

ds+− d

ds−

);

Vµ =d′

dsxµ = V µ − iUµ =

vµ+ + vµ

−2

− ivµ+ − vµ

−2

For the fluctuations one has

(4.12) LS < dξµ±dξν

± >= ∓2Dηµνds

One chooses here (+,−,−,−) for the Minkowski signature for ηµν and there is amild problem because the diffusion (Wiener) process makes sense only for posi-tive definite metrics. Various solutions were given in [314, 859, 1013] and theyare all basically equivalent, amounting to the transformatin a Laplacian into aD’Alembertian. Thus the two forward and backward differentials of f(x, s) shouldbe written as

(4.13) (df/ds±) = (∂s + vµ±∂µ ∓D∂µ∂µ)f

One considers now only stationary functions f, not depending explicitly on theproper time s, so that the complex covariant derivative operator reduces to

(4.14) (d′/ds) = (Vµ + iD∂µ)∂µ

Now assume that the large scale part of any mechanical system can be char-acterized by a complex action S leading one to write

(4.15) δS = −mcδ

∫ b

a

ds = 0; ds = LS <√

dXνdXν >

This leads to δS = −mc∫ b

aVνd(δxν) with δxν = LS < dXν >. Integrating by

parts yields

(4.16) δS = −[mcδxν ]ba + mc

∫ b

a

δxν(dVµ/ds)ds

To get the equations of motion one has to determine δS = 0 between the sametwo points, i.e. at the limits (δxν)a = (δxν)b = 0. From (4.16) one obtains thena differential geodesic equation dV/ds = 0. One can also write the elementaryvariation of the action as a functional of the coordinates. So consider the point aas fixed so (δxν)a = 0 and consider b as variable. The only admissable solutionsare those satisfying the equations of motion so the integral in (4.16) vanishes andwriting (δxν)b as δxν gives δS = −mcVνδxν (the minus sign comes from the choiceof signature). The complex momentum is now

(4.17) Pν = mcVν = −∂νS

and the complex action completely characterizes the dynamical state of the parti-cle. Hence introduce a wave function ψ = exp(iS/S0) and via (4.17) one gets

(4.18) Vν = (iS0/mc)∂ν log(ψ)

Now for the scale relativistic prescription replace the derivative in d/ds by itscovariant expression d′/ds. Using (4.18) one transforms dV/ds = 0 into

(4.19) − S20

m2c2∂µlog(ψ)∂µ∂ν log(ψ)− S0D

mc∂µ∂µ∂ν log(ψ) = 0


The choice S0 = = 2mcD allows a simplification of (4.19) when one uses theidentity

(4.20)12

(∂µ∂µψ

ψ

)=(

∂µlog(ψ) +12∂µ

)∂µ∂ν log(ψ)

Dividing byD2 one obtains the equation of motion for the free particle ∂ν [∂µ∂µψ/ψ] =0. Therefore the KG equation (no electromagnetic field) is

(4.21) ∂µ∂µψ + (m2c2/2)ψ = 0

and this becomes an integral of motion of the free particle provided the integrationconstant is chosen in terms of a squared mass term m2c2/2. Thus the quantumbehavior described by this equation and the probabilistic interpretation given toψ is reduced here to the description of a free fall in a fractal spacetime, in analogywith Einstein’s general relativity. Moreover these equations are covariant since therelativistic quantum equation written in terms of d′/ds has the same form as theequation of a relativistic macroscopic and free particle using d/ds. One notes thatthe metric form of relativity, namely V µVµ = 1 is not conserved in QM and it isshown in [775] that the free particle KG equation expressed in terms of V leadsto a new equality

(4.22) VµVµ + 2iD∂µVµ = 1

In the scale relativistic framework this expression defines the metric that is inducedby the internal scale structures of the fractal spacetime. In the absence of anelectromagnetic field Vµ and S are related by (4.17) which can be writen as Vµ =−(1/mc)∂µS so (4.22) becomes

(4.23) ∂µS∂µS− 2imcD∂µ∂µS = m2c2

which is the new form taken by the Hamilton-Jacobi equation.

REMARK 3.4.1. We go back to [716, 775] now and repeat some of theirsteps in a perhaps more primitive but revealing form. Thus one omits the LSnotation and uses λ ∼ 2D; equations (4.8) - (4.14) and (4.11) are the same andone writes now d/ds for d′/ds. Then d/ds = Vµ∂µ + (iλ/2)∂µ∂µ plays the roleof a scale covariant derivative and one simply takes the equation of motion ofa free relativistic quantum particle to be given as (d/ds)Vν = 0, which can beinterpreted as the equations of free motion in a fractal spacetime or as geodesicequations. In fact now (d/ds)Vν = 0 leads directly to the KG equation uponwriting ψ = exp(iS/mcλ) and Pµ = −∂µS = mcVµ so that iS = mcλlog(ψ) andVµ = iλ∂µlog(ψ). Then

(4.24)(Vµ∂µ +

iλ

2∂µ∂µ

)∂ν log(ψ) = 0 = iλ

(∂µψ

ψ∂µ +

12∂µ∂µ

)∂ν log(ψ)

Now some identities are given in [775] for aid in calculation here, namely

(4.25)∂µψ

ψ∂µ

∂νψ

ψ=

∂µψ

ψ∂ν

(∂µψ

ψ

)=

=12∂ν

(∂µψ

ψ

∂µψ

ψ

); ∂µ

(∂µψ

ψ

)+

∂µψ

ψ

∂µψ

ψ=

∂µ∂µψ

ψ


The first term in the last equation of (4.24) is then (1/2)[(∂µψ/ψ)(∂µψ/ψ)] andthe second is

(4.26) (1/2)∂µ∂µ∂ν log(ψ) = (1/2)∂µ∂ν∂µlog(ψ) =

= (1/2)∂ν∂µ∂µlog(ψ) = (1/2)∂ν

(∂µ∂µψ

ψ− ∂µψ∂µψ

ψ2

)Combining we get (1/2)∂ν(∂µ∂µψ/ψ) = 0 which integrates then to a KG equation

(4.27) −(2/m2c2)∂µ∂µψ = ψ

for suitable choice of integration constant (note /mc is the Compton wave length).

Now in this context or above we refer back to Section 2.2 for example and writeQ = −(1/2m)(R/R) (cf. Section 2.2 before Remark 2.2.1 and take = c = 1 forconvenience here). Then recall ψ = exp(iS/mλ) and Pµ = mVµ = −∂µS withiS = mλlog(ψ). Also Vµ = −(1/m)∂µS = iλ∂µlog(ψ) with ψ = Rexp(iS/mλ) solog(ψ) = iS/mλ = log(R) + iS/mλ, leading to

(4.28) Vµ = iλ[∂µlog(R) + (i/mλ)∂µS] = − 1m

∂µS + iλ∂µlog(R) = Vµ + iUµ

Then = ∂µ∂µ and Uµ = λ∂µlog(R) leads to

(4.29) ∂µUµ = λ∂µ∂µlog(R) = λlog(R)

Further ∂µ∂ν log(R) = (∂µ∂νR/R)− (RνRµ/R2) so

(4.30) log(R) = ∂µ∂µlog(R) = (R/R)− (∑

R2µ/R2) =

= (R/R)−∑

(∂µR/R)2 = (R/R)− |U |2

for |U |2 =∑

U2µ. Hence via λ = 1/2m for example one has

(4.31) Q = −(1/2m)(R/R) = − 12m

[|U |2 +

1λ

log(R)]

=

= − 12m

[|U |2 +

1λ

∂µUµ

]= − 1

2m|U |2 − 1

2div(U)

(cf. Section 2.2).

REMARK 3.4.2. The words fractal spacetime as used in the scale relativitymethods of Nottalle et al for producing geodesic equations (SE or KG equation) aresomewhat misleading in that essentially one is only looking at continuous nondif-ferentiable paths for example. Scaling as such is of course considered extensivelyat other times. It would be nice to create a fractal derivative based on scalingproperties and H-dimension alone for example which would permit the powerfultechniques of calculus to be used in a fractal context. There has been of coursesome work in this direction already in e.g. [187, 257, 411, 437, 466, 471, 562,721, 748, 816].


5. QUANTUM MEASUREMENT AND GEOMETRY

We consider here a paper [989], which is based in part on a famous paper ofLondon [611] (reprinted in [731]). In [611] it was shown that the ratio of the Weylscale factor to the Schrodinger wave function is constant if the proportionality con-stant between the Weyl potential and the EM potential is taken to be imaginary;this observation gave birth to modern gauge theories and the original Weyl theorywas absorbed into QM with the original scale freedom becoming invariance underunitary gauge transformations (cf. also Section 3.5.1). Both the Weyl theory andthe Schrodinger theory describe the evolution of a field in time and given the fac-tor of i and the Kaluza-Klein framework used by London, those evolutions are thesame. In the Weyl picture the field characterizes the length scales of fundamentalmatter, while in the Schrodinger picture it is the wave function corresponding to afundamental particle. This analogy is pursued further in [989] with a main themebeing the equivalence between Weyl measurement and quantum measurement; acomplete theory of measurement in a Weyl geometry is said to contain the crucialelements of quantization and analogies of the following sort are indicated.

(5.1)

Weyl − quantum correspondence Quantum mechanicsZero−Weyl − weight number Real eigenvalue

Diffusion equation SEWeiner path integral Feynman path integral

Weightful length field ψw Complex state function ψWeyl conjugate ψ−w ψ∗

Probability ψwψ−w Probability |ψ|2ψw → ewφψw (conformal) ψ → eiφψ (unitary)

We will try to make sense out of this following [989] (cf. also [63, 64]). Begin witha real 4-dimensional manifold (M, [g]) where [g] is a conformal equivalence classof Lorentz metrics. In addition to local coordinate transformations one has Weyl(conformal) transformations given via T (x)′ = exp[w(T )Λ(x)]T (x) where T is atensor field and w(T ) is the Weyl weight (a real number). One takes a coordinatebasis Eα = ∂/∂xα and Eα = dxα in the tangent and cotangent space satisfyingw(Eα) = w(Eα) = 0.

DEFINITION 5.1. One defines a torsion free derivative D via• Linearity: D(aT1 + bT2) = aDT1 + bDT2 for real a, b• Leibniz: D(T1T2) = (DT1)T2 + T1(DT2)• Weyl covariant: D(fT ) = [df + w(f)Wf ]T + fDT where W is a real

1-form (Weyl potential)• Zero weight: w(DT ) = w(T )

Under a Weyl transformation W → W ′ = W − dΛ and one has

(5.2) DT = DµTαβEµ ⊗Eα ⊗ Eβ ; DµTα

β =

= ∂µTαβ + T ρ

βΓαρµ − Tα

ρΓρβµ + w(T )WµTα

β

There is no unique metric on the space; instead the metric is to be taken ofthe Weyl type w(g) = 2 so that under a Weyl transformation g′ = exp[2Λ(x)]g.

5. QUANTUM MEASUREMENT AND GEOMETRY 133

The principle fields of the theory are related by the requirement Dg = 0, or incomponents

(5.3) Dµgαβ = 0 = ∂µgαβ − gρβΓραµ − gαρΓ

ρβµ + 2Wµgαβ

This can be solved to give

(5.4) Γαβµ =

α

β µ

+ (δα

βWµ + δαµWβ − gβµWα)

Vanishing torsion has been assumed in (5.4) so that the bracket expression is theusual Christoffel connection. The curvature tensor is then

(5.5) Rαβµν = Γα

βν,µ − Γαβµ,ν + Γα

ρµΓρβν − Γα

ρνΓρβµ

Unlike the Riemannian curvature tensor the Weyl curvature has nonvanishing traceon the first pair of indices so that (1/2)Rα

αµν = Wν,µ−Wµ,ν = Wµν where Wµµ isthe gauge invariant field strength of the Weyl potential. One says that two fieldsare Weyl conjugate if they have the same Lorenz transformation properties butopposite Weyl weights.

Now for a theory of measurement one first looks at zero weight fields. In thisdirection note that fields with nonvanishing Weyl weight will experience changesunder parallel transport. For example the mass squared transported along a pathwith unit tangent vector uµ = dxµ/dτ satisfies

(5.6) 0 = uµDµ(m2) = uµ∂µ(m2) + w(m2)uµWµm2

Integrating along the path of motion one finds a path dependence of the formm2 = m2

0exp[w(m2)∫

Wµuµdτ ] where the line integral has been written in termsof the path parameter τ . Note this is analogous to m2 = m2

0exp(Q) in theShojai theory of Section 3.2 suggesting some relation to a quantum potentialQ ∼ w(m2)

∫Wµuµdτ . However at this point there is no quantum matter posited

and no density ρ so a Weyl vector Wµ ∼ ∂µlog(ρ) as in Remark 3.3.1 is unten-able and no comparison to (3.28) can be undertaken. However this does show ageometrical dependence of mass in general and in the flat space of Remark 3.3.1it is replaced by a quantum potential. Indeed this (Schouten-Haantjes) confor-mal mass thus depends on the Weyl vector and if two particles of identical massare allowed to propagate freely (by parallel transport) along different paths andbrought together there will be a mass difference

(5.7) ∆m2 = m20e

w(m2)∮

Wµuµdτ ≡ m20e

w(m2)∫

SWµνdSµν

where dSµν is an element of any 2-surface S bounded by the closed curve definedby the two particles. Hence unless the surface integral of the Weyl field strengthvanishes there will be a path dependence for masses and of any other field ofnonzero weight. One postulates now (I) that all quantities of vanishing Weylweight should be physically meaningful (observables) and (II) that all fields occurin conjugate pairs satisfying conjugate equations of motion. Assume that M±evolves by parallel transport along a path as above via

(5.8) 0 = uµDµD± = uµDµM± ± w(M)M±Wµuµ

where D is a derivation using the full connection (5.4) and one sets w(M) =w(M+) > 0 for convenience. One has also

(5.9) M± = Mexp[∓w(M)∫

Wµuµdτ ]

where M is weightless with uµDµM = 0. Now suppose one wants to measure somecharacteristic of M (i.e. of M+ or M−). M can be scaled by an arbitrary gaugefunction and one transports M along a path so that its covariant derivative in thedirction of motion vanishes. Then the change in size is specified by (5.9) but itis not clear that we can tell what path a particle has taken. In a Riemannianspace there are geodesics determining the paths of classical matter but that isnot true in a Weyl space (in this regard we refer to [188], Section 3.2, and to[133, 796, 797, 798, 799]).

In order to study the motion of M one begins with the observation that a Weylgeometry provides a probability PAB(M) of finding a value M at a point B for asystem which is known to have had a value M0 at point A. Finding PAB(M) istantamount to finding the fraction of paths which the system may follow leadingto any given value of M. Since there may be no special paths in a Weyl geometryone has to settle for moments of the distribution. To find the average value ofmagnitude of M denoted by < M > one integrates (5.9) over all paths via

(5.10) < M >=∫D[x]M0exp[w(M)

∫ B

A

Wµuµdτ ]

where the usual path integral normalization is included implicitly in D[x] (see e.g.[362, 457, 855]) for path integrals). However this gives no information as towhether one should expect M to actualy reach B. In [989] there is then a longdiscussion (and a detailed Appendix) involving path averages, probability, Wienerintegrals, etc. plus a postulate (III) that the probability a system will undergoa given infinitesimal displacement xµ is inversely proportional to the change inlength such a displacement produces in the system. Now d = w(M)Wµdxµ =w(M)Wµuµdτ and a plausible (rigorous) argument is given then to represent theprobability of the system reaching any spacetime point x from x0 as

(5.11) G(x0;x) =∫D[x]exp[w(M)

∫ x

x0

Wµuµdτ ]

(which bears an obvious resemblence to (5.10)). Comparison of (5.10) and (5.11)involves noting first that (5.11) is gauge dependent but the gauge dependencecomes out of the path integral since it depends only on the end points. Thus

(5.12) G′(x0;x) =∫D[x]exp[w(M)

∫ x

x0

(Wµ − ∂µφ)uµdτ ] =

= e−w(M)[φ(x)−φ(x0)]

∫D[x]exp[w(M)

∫ x

x0

Wµuµdτ ]


This means that one can eleiminate the gauge factor by multiplying by the Weylconjugate expression

(5.13) G′(x0;x) = ew(M)[φ(x)−φ(x0)]

∫D[x]exp[−w(M)

∫ x

x0

Wµuµdτ ]

to give a meaningful gauge invariant probability P (x0, x) = G(x0;x)G(x0;x) whichis the probability of detecting the dilating system at x given its presence at x0. Itmay be thought of as the joint probability of finding both M and M at x. Hereone is dealing with a real path integral, unlike QM, and the phase invariance of awave function ψ′ = exp(iφ)ψ is replaced by conformal invariance M ′ = exp(φ)M(this is the same factor of i introduced by London in 1927). Since that timegauge transformations have appeared as phases and the wave interpretation hasbeen maintained; now one maintains a real gauge transformation and changesthe interpretation of physical phenomena (see [989] for more discussion in thisdirection).

Now one shows the equivalence to QM of the nonrelativistic limit of (5.11)when the exponent in the path integral is identified with a multiple of the classicalaction, i.e.

∫Wµuµdτ = λS = λ

∫Ldτ . The integrands here may also be equated

except for the possible addition of the total derivative of a function of τ . Butsuch a derivative is already known to be both a gauge freedom of Wµ and atransformation of L that leaves the equations of motion unaltered. So the possibleequivalent versions of L may be understood as gauge changes of the underlyinggeometry. This identification fixes the physical interpretation of Wµ up to thegauge choice and since uµ = xµ equating the integrands gives

(5.14) λPµ = λ(∂L/∂uµ) = Wµ

so that Wµ is proportional to the generalized momentum Pµ conjugate to xµ.Now Weyl had originally identified Wµ with the derivative of an EM potential∂µU ∼ Aµ and the present approach suggests Wµ = λ(pµ + Aµ) so that all energyprovides a surce of expansion rather than just EM energy. This still allows gaugetransformations of Wµ to be identified with gauge transformations of Aµ. Next onegoes to the nonrelativistic limit of the path integral to find a differential equationfor the amplitudes G(x0;x). It is convenient to explicitly separate the kineticterm pµuµ from Wµuµ which will enable one to identify the path integral in (5.11)with a Wiener integral. Thus with full generality one writes Wµ = λ(pµ + Wµ)where any gauge transformation is understood to apply to Wµ. Now consider thenonrelativistic limit where the integral

∫pµuµdτ ∼ mc2

∫dτ so that mc2

∫dτ ∼∫

[mc2 − (m/2)v2]dt. To this order the path integral becomes (suppressing limitsof integration)

(5.15) G(x0;x) =∫D[x]eλw(M)

∫[(1/2)mv2+W ·v−W 0−mc2)]dt

This is of the form

(5.16) P (x0;x) =∫D[x]exp[−(1/2)

∫((q + w)2 −∇ ·w)dt]


where P (x0;x) is the propagator for the Fokker-Planck equation ∂tP = (1/2)∇2P+∇ · (wP ) provided one makes the identifications

(5.17) q =√−w(M)λmv; ∇x =

√−w(M)λm∇q; ψ = Pe−2mc2

;

w =√−w(M)λ/mW; 2w(M)λ(mc2 + W0) = w2 −∇ ·w

(cf. [457, 672, 674, 698, 856]). Carrying out the substitutions and settingλWµ = −U(λφ,A) one obtains ψ(x) =

∫ψ(x′)G(x, x′)dx′ as a solution to

(5.18)1

w(M)λ∂tψ = − 1

2m[w(M)λ]2[∇+ w(M)λA]2 + (mc2 + Uφ)ψ

with initial condition ψ = ψ(x′) (this should be checked to clarify the roles of Uand φ). If one sets λ = −1 and the time is allowed to become imaginary theSE minimally coupled to EM arises. Thus choose λ = −1 but leave time alonesince it is not needed; then (5.18) can be interpreted as a stochastic form of QM.Evidently the Weyl weight serves the function of i, changing sign appropriately forthe conjugate field. The emergence of the Fokker-Planck equation indicates diffu-sion and this is discussed at length in [186, 672, 674, 698, 856]. In addition thematter is discussed in [989] from various points of view. In particular one takes(1/)S =

∫Wµuµdτ and observes that a classical limit of the Weyl geometry will

exist whenever there is an extremum to the action (as in the Feynman path inte-gral). Thus a classical limit of (5.11) occurs whenever Ψ = exp[w(M)

∫ x

x0Wµuµdτ ]

is extremal. However there is a difference here involving Ψ as a length factor.One shows that δΨ = 0 corresponds to a special case of the Weyl field since∫ B

Adτ(Wµ,ν−Wν,µ)uµδxν = 0 arises via variation which means Wµνuν = 0. Some

calculation then shows that Wα = ξ∂αχ (up to a gauge transformation) for anyappropriately normalized functions ξ, χ satisfying

(5.19) (Dµχ)uν = (Dµχ)vµ = (Dµξ)uν = (Dµξ)vµ = 0;

(1/2)εµναβWαβ = uµvν − uνvµ

with ε the Levi-Civita tensor (cf. [302, 989]). Now Wα = ξ∂αχ is a ratherremarkable relation; it represents a restricted form of Wα since it is easy to finda Weyl vector such that Wµνuν ∼ Wµ0 = 0 for all nonspacelike uν . Since thisformula arises for an arbitrary set of paths uα it is clear that not all Weyl fieldswill have a classical limit. Thus as argued at the beginning the generic Weylgeometry lacks preferred paths and requires a path average. On the other handif one chooses a gauge where Wαuα = 0 (which is possible) then weightful bodiesfollowed the preferred classical trajectories and experience no dilation. There isconsiderable discussion along these lines in [989] which is omitted here; thereis also interesting material on relations to general relativity. In particular it ispointed out that size changes associated with nonvanishing Weyl field strengthare not necessarily classically observable. However the Weyl field itself must bepresent and consequently must be detectable. Finding the physical field that itcorresponds to simply requires substituting the appropriate conjugate momentumfor Wµ in the classical equation of motion Wµνuν = 0. Since the only long rangeforces are gravity and EM and gravity is still accounted for by the Riemanniancurvature, Wµ must be electromagnetic. The most general classical conjugate


momentum is therefore that of a point particle with charge q moving in an EMfield. Then in an arbitrary gauge

(5.20) Wµ = (1/)(pµ + qAµ + ∂µΛ)

where pµ = muµ and uµuµ = −1. Then

(5.21) 0 = Wµνuν = (1/)(pµ,ν − pν,µ + qAµ,ν − qAνµ)uν

or (using (uµuµ),ν = 0) dpµ/dτ = quνFµν which is the Lorenz force law (note thatPlanck’s constant drops out). For the interpretation of Wµ itself one can combinethe curl of (5.20) with

(5.22) Wαβ = DαχDβξ −DβχDαξ = ∂αχ∂βξ − ∂βχ∂αξ

(cf. (5.19) and the surrounding discussion); this leads to

(5.23) ∂αχ∂βξ − ∂βχ∂αξ = (1/)(pα,β − pβ,α + qAα,β − qAβ,α)

the time component of which gives again the Lorenz law. The spatial componentscan be solved for the magnetic field to give

(5.24) B = (/q)(∇χ×∇ξ)− (m/q)(∇× v)

The two fields χ and ξ on the right side of B are sufficient to guarantee theexistence of any type of physical magnetic field. Conversely one can use (5.24)to solve for the Weyl field in terms of B and v (which of course depend on ).One notes that for vanishing Weyl field (5.24) reduces to the London equationfor superconductivity. This means that matter fields which conspire to produce aRiemannian geometry become superconducting.

5.1. MEASUREMENT ON A BICONFORMAL SPACE. We con-tinue the theme of Section 3.5 with a more general perspective from [35] based onbiconformal geometry (cf. Appendix E for some background material and see also[35, 36, 113, 497, 558, 987, 980, 981, 989, 990, 991, 992, 993, 994, 1010]).We regard this approach via biconformal geometry as very interesting and will tryto present it faithfully. The background material in Appendix E should be readfirst; results in [994] for example create a unified geometrical theory of gravityand electromagnetism based on biconformal geometry. One develops in [35] an in-terpretation for quantum behavior within the context of biconformal gauge theorybased on the following postulates:

(1) A σC biconformal space provides the physical arena for quantum andclassical physics.

(2) Quantities of vanishing conformal weight comprise the class of physicallymeaningful observables.

(3) The probability that a system will follow any given infinitesimal dis-placement is inversely proportional to the dilatation the displacementproduces in the system.

From these assumptions follow the basic properties of classical and quantum me-chanics. The symplectic structure of biconformal space is similar to classical phasespace and also gives rise to Hamilton’s equations, Hamilton’s principal function,


conjugate variables, fundamental Poisson brackets, and Liouville theory when pos-tulate 3 is replaced by a postulate of extremal motion. We sketch this here (some-what brutally) and refer to [35] for details, philosophy, and further references;the details for the biconformal geometry are spelled out in [992, 994]. Thus onewants a physical arena which contains 4-D spacetime in a straightforward mannerbut which is large enough and structured so as to contain both general relativ-ity (GR) and quantum theory (QT) at the same time. One demands thereforeinvariance under global Lorentz transformations, translations, and scalings (seebelow) and the Lie group characterizing this is the conformal group O(4, 2) or itscovering group SU(2, 2). In Appendix E the basic facts about Lorentz transforma-tions Ma

b = −Mba = ηacMcb, translations Pa, special conformal transformations

Ka, and dilatations D are exhibited in the context of conformal gauge theory(a, b = 0, 1, 2, 3). One has two involutive automorphisms of the conformal algebra,first

(5.25) σ1 : (Mab, Pa,Ka, D) → (Ma

b,−Pa,−Ka, D)

which identifies the residual local Lorentz and dilatation symmetry characteristicof biconformal gauging and this corresponds (resp. for the Poincare Lie algebraor the Weyl algebra) to

(5.26) σ1 : (Mab, Pa)→ (Ma

b,−Pa) or σ1 : (Mab, Pa, D) → (Ma

b,−Pa, D)

There is also a second involution for the conformal group, namely

(5.27) σ2 : (Mab, Pa,Ka, D) → (Ma

b,Ka, P a,−D)

Some representations of the conformal algebra, namely su(2, 2), are necessarilycomplex and σ2 can be realized as complex conjugation. Specifically one thinks ofa representation in which Pa and Ka are complex conjugates while Ma

b is real andD is purely imaginary and such representations will be called σC representations.Biconformal spaces for which the connection 1-forms (and hence curvatures) havethis property are then called σC spaces (see Appendix E for examples). This leadsto postulate 1 above, namely the physical arena for QT and classical physics is aσC biconformal space. Now biconformal gauging of the conformal group providesin particular a symplectic structure as follows. Gauging D introduces a singlegauge 1-form ω (the Weyl vector) and the corresponding dilatational curvature2-form is

(5.28) Ω = dω − 2ωaωa

where ωa, ωa are 1-form gauge fields for the translation and special conformaltransformations respectively, which span an 8-dimensional space as an orthonormalbasis (note ωa = ηabω

b for σC representations and products are wedge products).Now for all torsion free solutions to the biconformal field equations (i.e. ∗d∗dω0

0 =J, ω0

a = Ta + ·, etc. - cf. Appendix E) the dilatational curvature takes the form(•) Ω = κωaωa with κ constant, so the structure equation becomes (••) dω = (κ+2)ωaωa. As a result dω is closed and nondegenerate and hence symplectic (sinceωa, ωa span the space). There is also a biconformal metric arising from the groupinvariant Killing metric KΣΠ = cΛ

∆Σc∆ΛΠ where cΛ

∆Σ (Σ,Π, · · · = 1, 2, · · · , 15) arethe real structure constants from the Lie algebra. This metric has a nondegenerate


projection to the 8-D subspace spanned by Pa, Ka and provides a natural pseudo-Riemannian metric on biconformal manifolds. The projection takes the form

(5.29) KAB =(

ηab

ηab

)(A,B = 0, 1, · · · , 7)

One defines now conformal weights w of a definite weight field F via () Dφ :F → [exp(wφ)]F where Dφ is dilatation by exp(φ) (cf. [989] and Section 3.5).One assumes now postulate 2 and concludes that for a field with nontrivial Weylweight to have physical meaning it must be possible to construct weightless scalarsby combining it with other fields (easily done with conjugate fields); one notesthat zero weight fields are self conjugate. The symplectic form Θ = ωaωa definesa symplectic bracket via

(5.30) f, g = ΘMN ∂f

∂uM

∂g

∂uN

where uM = (xa, yb). For real solutions f, g to the field equations f and g areconjugate if they satisfy f, f = 1, f, f = g, g = 0. However for σC rep-resentations ω is a pure imaginary 1-form since it is defined as the dual to thedilatation generator D which is pure imaginary. One sees then that

(5.31) ωaωa = ωaωa = ηabωbηacωc = −ωaωa

so the dilatational curvature and the symplectic form are imaginary (cf. also[35, 529]). Consequently, for use of a complex gauge vector with real gaugetransformations, the fundamental brackets should take here the form

(5.32) f, g = i; f, f = g, g = 0; wf = −wg

In an arbitrary biconformal space one sets either

(5.33)1S =

1

∫Ldλ =

∫ω =

∫(Wadxa + Wadya) or

i

S =

i

∫Ldλ =

∫ω =

∫(Wadxa + Wadya)

The second form holds in a σC representation for the conformal group. An ar-bitrary parameter λ is OK since the integral of the Weyl 1-form is independentof parametrization. This integral also governs measurable size change since underparallel transport the Minkowski length of a vector V a changes by

(5.34) = 0exp

∫ω; 2 = ηabV

aV b

(cf. Appendix E). This change occurs because ηab = (−1, 1, 1, 1) is not a naturalstructure for biconformal space. This is in contrast to the Killing metric KAB

where lengths are of zero conformal weight. In a σC representation the Weyl vectoris imaginary so the measurable part of the change in is not a real dilatation -rather, it is a change of phase. Now for classical mechanics one uses a variation ofpostulate 3, namely: The motion of a (classical) physical system is givenby extrema of the integral of the Weyl vector. Biconformal spaces are realsymplectic manifolds so the Weyl vector can be chosen so that the symplectic form


satisfies the Darboux theorem ω = Wadxa = −yzdxa; for σC representations theDarboux equations still holds but now with

(5.35) ω = Wadxa = −iyadxa

and the classical motion is independent of which form is chosen. Thus the symplec-tic form for the σC case is Θ = dω = −idyadxa and one has () xa, yb = iδa

b .Thus from () it follows that yb is the conjugate variable to the position coordi-nate xb and in mechanical units one may set ya = αpa with

(5.36) iαS =∫

ω = −iα

∫(p0dt + pidxi)

(α can be any constant with appropriate dimensions). Now if one requires t as aninvariant parameter (so δt = 0) one can vary the corresponding canonical bracketto find

(5.37) 0 = δt, p0 = δt, p0+ t, δp0 =∂(δp0)∂p0

Thus δp0 can depend only on the remaining coordinates so δp0 = −δH(yi, xj , t)

and the existence of a Hamiltonian is a consequence of choosing time as a nonvariedparameter of the motion. Applying the postulate δS = 0 variation leads to

(5.38) 0 = iαδS = −iα

∫(δp0dt + δpidxi − dpiδx

i) =

= −iα

∫ (−∂H

∂xiδxidt− ∂H

∂piδpidt + δpidxi − dpiδx

i

)and this gives the standard Hamilton’s equations

(5.39) 0 = −∂H

∂pidt + dxi; 0 = −∂H

∂xidt− dpi

(note i and α drop out of the equations).

In the presence of nonvanishing dilatational curvature one then considers aclassical experiment to measure size (or phase) change along C1, while a rulermeasured by λ moves along C2 (Ci are classical paths between two fixed points).Some argument (see [35]) leads to an unchanged ratio of lengths via

(5.40)

λ=

0λ0

exp

∫C−1−C−2

ω =0λ0

exp

∮ω =

0λ0

exp

∫ ∫S

dω =0λ0

where S is any surface bounded by the closed curve C1−C2 (cf. also Section 3.5).Thus no dilatations are observable along classical paths. This calculation alsoshows that the restriction of ω to classical paths is exact and proves the existenceof Hamilton’s principal function S with

(5.41) αS(x) =∫ x

Wadxa =∫ x

Wadxa

dtdt

There is further argument in [35] via gauge freedom to show that classical objectsdo not exhibit measurable length change (in the complex case the phase changescannot be removed by gauge choice but they are unobservable). Relations betweenphase space and biconformal space are discussed and one arrives at QM.


From the above one knows that there is no measurable size change alongclassical pathes in a biconformal geometry but for systems evolving along otherthan extremal paths (where the Hamilton equations do not apply for example)there may be measurable dilatation. To deal with this one needs nonclassicalmotion and one goes to the basic postulate 3, namely that the probability a systemwill follow any given infinitesimal displacement is inversely proportional to thedilatation the displacement produces in the system. The properties of biconformalspace determine the evolution of Minkowski lengths along arbitrary curves andthe imaginary Weyl vector produces measurable phase changes in the same wayas the wave function. Combining this with the classically probabilistic motion ofpostulate 3, together with the necessary use of a standard of length to complywith postulate 2, one concludes that the probability of a system at xa

0 arriving atthe point xa

1 is given by

(5.42) P (xi1) =

∫D[xC′ ]exp

(∫C′

ω

)∫D[xC ]exp

(−∫

C

ω

)=

= P(xi1)P(−xi

1) = P(xi1)P(xi

1)

where a path average over all paths connecting the two points is involved and P(x)is simulaneously the probability amplitude of the conformally conjugate systemreaching xi

1. Here one considers ratios /λ as above and includes all possibleruler paths. These are standard Feynman path integrals which are known tolead to the Schrodinger equation (not Wiener integrals as in [989]) and it is therequirement of a length standard that forces the product structure in (5.42). Notethat the phase invariance of a wave function ψ′ = exp(iφ)ψ is created by the σC

conformal invariance M ′ = exp(λw)M . The i in the Weyl vector is the crucial inoted by London in [611] (cf. [989] and Section 3.5). Note also that the pathintegral in (5.42) and the biconformal paths depend generically on the spacetimeand momentum variables so one can immediately generalize to the usual integralsof QM, namely

(5.43) P(xi1) =

∫D[xC ]D[yC ]exp(

∫Cω)

Note also that the failure of the base space to break into space like and mo-mentum like submanifolds indicates a fundamental coupling between position andmomentum and suggests a connection to the Heisenberg uncertainty principle.The arguments in [35] have a somewhat heuristic flavor at times but are certainlyplausible and do refine the techniques of [989] (sketched in Section 3.5) in manyways. Given the success of biconformal geometry in unifying GR and EM it wouldseem only natural and just that QM could be encompassed as well in the sameframework and further developments are eagerly awaited.

REMARK 3.5.1 We note from [993] that when identifying biconformal coor-dinates (xµ, yν) with phase space coordinates (xµ, pν) one sets naturally yν = βpν .This β must account for a sign difference in ηµνβpµβpν = −ηµνyµyν (cf. [993]) soβ is pure imaginary. Further to account for the different units of yν (length−1) andpν (momentum) one chooses yν = (i/)pν and this relation between the geometric


variables of conformal gauge theory and the physical momentum variables is thesource of complex quantities in QM.

CHAPTER 4

GEOMETRY AND COSMOLOGY

This chapter and the next will cover a number of more or less related topicshaving to do with cosmology, the zero point field (ZPF), the aether and vacuum,quantum geometry, electromagnetic (EM) phenomena, and Dirac-Weyl geometry.

1. DIRAC-WEYL GEOMETRY

A sketch of Dirac Weyl geometry following [302] was given in [188] in con-nection with deBroglie-Bohm theory in the spirit of the Tehran school (cf. [117,118, 668, 669, 831, 832, 833, 834, 835, 836, 837, 838, 864, 865, 866, 867,868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879]). We go now to[498, 499, 500, 501, 502, 503, 817] for generalizations of the Dirac Weyl the-ory involved in discussing magnetic monopoles, dark matter, quintessence, mattercreation, etc. We skip [500] where some notational problems seem to arise in theLagrangian and go to [499] where in particular an integrable Weyl-Dirac theory isdeveloped (the book [498] is a lovely exposition but the work in [499] is somwhatnewer). Note, as remarked in [645] (where twistors are used), the integrable Weyl-Dirac geometry is desirable in order that the natural frequency of an atom at apoint should not depend on the whole world line of the atom. The first paperin [499] is designed to investigate the integrable Weyl-Dirac (Int-W-D) geometryand its ability to create massive matter. For example in this theory a sphericallysymmetric static geometric formation can be spatially confined and an exteriorobserver will recognize it as a massive entity. This may be either a fundamen-tal particle or a cosmic black hole both confined by a Schwarzschild surface. Wesummarize again some basic features in order to establish notation, etc. Thus inthe Weyl geometry one has a metric gµν = gνµ and a length connection vector wµ

along with an idea of Weyl gauge transformation (WGT)

(1.1) gµν → gµν = e2λgµν ; gµν → gµν = e−2λgµν

where λ(xµ) is an arbitrary differerentiable function (cf. also [319, 320, 762,853, 854] for Weyl geometry). One is interested in covariant quantities satisfyingψ → ψ = exp(nλ)ψ where the Weyl power n is described via π(ψ) = n, π(gµν) = 2,and π(gµν) = −2. If n = 0 the quantity ψ is said to be gauge invariant (in-invariant). Under parallel displacement one has length changes and for a vector

(1.2) (i) dBµ = −BσΓµσνdxν ; (ii) B = (BµBνgµν)1/2; (iii) dB = Bwνdxν

143

144 4. GEOMETRY AND COSMOLOGY

(note π(B) = 1). In order to have agreement between (i) and (iii) one requires

(1.3) Γλµν =

λ

µ ν

+ gµνwλ − δλ

ν wµ − δλµwν

where

λµ ν

is the Christoffel symbol based on gµν . In order for (iii) to hold in

any gauge one must have the WGT wµ → wµ = wµ + ∂µλ and if the vector Bµ istransported by parallel displacement around an infinitesimal closed parallelogramone finds

(1.4) ∆Bλ = BσKλσµνdxµδxν ; ∆B = BWµνdxµδxν

where

(1.5) Kλσµν = −Γλ

σµ,ν + Γλσν,µ − Γα

σµΓλαν + Γα

σνΓλαµ

is the curvature tensor formed from (1.3) and Wµν = wµ,ν − wν,µ. Equationsfor the WGT wµ → wµ and the definition of Wµν led Weyl to identify wµ withthe potential vector and Wµν with the EM field strength; he used a variationalprinciple δI = 0 with I =

∫L√−gd4x with L built up from Kλ

σµν and Wµν . Inorder to have an action invariant under both coordinate transformations and WGThe was forced to use R2 (R the Riemannian curvature scalar) and this led to thegravitational field.

Dirac revised this with a scalar field β(xν) which under WGT changes viaβ → β = e−λβ (i.e. π(β) = −1). His in-invariant action integral is then (f,µ ≡∂µf)

(1.6) I =∫

[WλσWλσ − β2R + β2(k − 6)wσwσ + 2(k − 6)βwσβ,σ+

+kβ,σβ,σ + 2Λβ4 + LM ]√−gd4x

Here k is a parameter, Λ is the cosmological constant, LM is the Lagrangian densityof matter, and an underlined index is to be raised with gµν . Now according to(1.4) this is a nonintegrable geometry but there may be situations when geometricvector fields are ruled out by physical constraints (e.g. the FRW universe). Inthis case one can preserve the WD character of the spacetime by assuming thatwν is the gradient of a scalar function w so that wν = w,ν = ∂νw. One has thenWµν = 0 and from (1.4) results ∆B = 0 yielding an integrable spacetime (Int-W-Dspacetime). To develop this begin with (1.6) but with wν given by wν = ∂νw sothe first term in (1.6) vanishes. The parameter k is not fixed and the dynamicalvariables are gµν , w, and β. Further it is assumed that LM depends on (gµν , w, β).For convenience write

(1.7) bµ = (log(β)),µ = β,µ/β

and use a modified Weyl connection vector Wµ = wµ+bµ which is a gauge invariantgradient vector. Write also k − 6 = 16πκ and varying w in (1.6) one gets a fieldequation

(1.8) 2(κβ2W ν);ν = S

1. DIRAC-WEYL GEOMETRY 145

where the semicolon denotes covariant differentiation with the Christoffel symbolsand S is the Weylian scalar charge given by 16πS = δLM/δw. Varying gµν onegets also

(1.9) Gνµ = −8π

T νµ

β2+ 16πκ

(W νWµ −

12δνµW σWσ

)+

+2(δνµbσ

;σ − bν;µ) + 2bνbµ + δν

µbσσ − δν

µβ2Λwhere Gν

µ represents the Einstein tensor and the EM density tensor of ordinarymatter is

(1.10) 8π√−gTµν = δ(

√−gLM )/δgµν

Finally the variation with respect to β gives an equation for the β field

(1.11) R + k(bσ;σ + bσbσ) = 16πκ(wσwσ − wσ

;σ) + 4β2Λ + 8πβ−1B

Note in (1.11) R is the Riemannian curvature scalar and the Dirac charge B is aconjugate of the Dirac gauge function β, namely 16πB = δLM/δβ.

By a simple procedure (cf. [302]) one can derive conservation laws; considere.g. IM =

∫LM

√−gd4x. This is an in-invariant so its variation due to coordinatetransformation or WGT vanishes. Making use of 16πS = δLM/δw, (1.10), and16πB = δLM/δβ one can write

(1.12) δIM = 8π∫

(Tµνδgµν + 2Sδw + 2Bδβ)√−gd4x

Via xµ → xµ = xµ + ηµ for an arbitrary infinitesimal vector ηµ one can write

(1.13) δgµν = gλνηλ;µ + gµληλ

;ν ; δw = w,νην ; δβ = β,νην

Taking into account xµ → xµ we have δIM = 0 and making use of (1.13) one getsfrom (1.12) the energy momentum relations

(1.14) Tλµ;λ − Swµ − βBbµ = 0

Further considering a WGT with infinitesimal λ(xµ) one has from (1.12) the equa-tion S+T−βB = 0 with T = T σ

σ . One can contract (1.9) and make use of (1.8) andS + T = βB giving again (1.11), so that (1.11) is a corollary rather than an inde-pendent equation and one is free to choose the gauge function β in accordance withthe gauge covariant nature of the theory. Going back to the energy-momentumrelations one inserts S + T = βB into (1.14) to get Tλ

µ;λ − Tbµ = SWµ. Now goback to the field equation (1.9) and introduce the EM density tensor of the Wµ

field

(1.15) 8πΘµν = 16πκβ2[(1/2)gµνWλWλ −WµW ν ]

Making use of (1.8) one can prove Θλµ;ν −Θbµ = −SWµ and using Tλ

µ;λ − TBµ =SWµ one has an equation for the joint energy momentum density

(1.16) (Tλµ + Θλ

µ);λ − (T + Θ)bµ = 0

One can derive now the equation of motion of a test particle (following [817]).Consider matter consisting of identical particles with rest mass m and Weyl scalarcharge qs, being in the stage of a pressureless gas so that the EM density tensor can


be written Tµν = ρUµUν where Uµ is the 4-velocity and the scalar mass densityρ is given by ρ = mρn with ρn the particle density. Taking into account theconservation of particle number one obtains from Tλ

µ;λ−Tbµ = SWµ the equationof motion

(1.17)dUµ

ds+

µλ σ

UλUσ =

(bλ +

qs

mWλ

)(gµλ − UµUλ)

In the Einstein gauge (β = 1) we are then left with

(1.18)dUµ

ds+

µλ σ

UλUσ =

qs

mwλ(gµλ − UµUλ)

EXAMPLE 1.1. Following [499] one considers a static spherically symmetricsituation with line element

(1.19) ds2 = eνdt2 − eλdr2 − r2(dθ2 + Sin2(θ)dφ2)

and all functions λ, ν, β, w, T νµ , S, B depend only on r. One looks for local

phenomena so Λ = 0. The field equations (1.9) can be written explicitly for(µν) = (0, 0), (1, 1), (2, 2), or (3, 3) to obtain

(1.20) e−λ

(−λ′

r+

1r2

)− 1

r2= −8πT 0

0

β2+

+2e−λ

(− (b′)2

2− b′′ +

λ′b′

2− 2b′

r

)+ 8πκe−λ(W ′)2;

e−λ

(ν′

r+

1r2

)− 1

r2= −8πT 1

1

β2− 2e−λ

(ν′b′

2+

2b′

r+

3(b′)2

2

)− 8πκe−λ(W ′)2;

14

(ν′′ +

(ν′)2

2+

ν′ − λ′

r− ν′λ′

2

)=

= −4eλπT 22

β2−(

b′′ +(ν′ − λ′)b′

2+

b′

r+

(b′)2

2

)+ 4πκe−λ(W ′)2

From (1.8) one has the equation for the W field

(1.21) 2κ[W ′′ +

(2b′ +

ν′ − λ′

2+

2r

)W ′

]= −eλS

β2

The most intriquing situation is when ordinary matter is absent, so Tµν = 0, and

then from S + T = βB and Tλµ;λ − Tbµ = SWµ one has S = 0 and B = 0. Take

first the simple case when W ′ = 0 or κ = 0 so (1.21) is satisfied identically and(1.20) takes the simple form

(1.22) e−λ

(−λ′

r+

1r2

)− 1

r2= 2e−λ

(− (b′)2

2− b′′ +

λ′b′

2− 2b′

r

);

e−λ

(ν′

r+

1r2

)− 1

r2= −2e−λ

(ν′b′

2+

2b′

r+

2(b′)2

2

);

e−λ

2

(ν′′ +

(ν′)2

2+

ν′ − λ′

r− ν′λ′

2

)= −2e−λ

(b′′ +

ν′ − λ′

2+

b′

r+

(b′)2

2

)Subtracting the first equation from the second one obtains (1/r)(λ′ + ν′) = 2b′ −(λ′ + ν′)b′ − 2(b′)2. The scalar b = log(β) is still arbitrary so that one can impose


a condition on it. Thus writing b′′ − (b′)2 = 0 one can integrate to get b(r) =log[1/(a− cr)] (curiously enough this is true) with a, c arbitrary constants whichare taken to be positive. Using b′′ = (b′)2 one obtains the equation λ′ + ν′ = 0and hence via b = log[1/(a − cr)] there rsults from (1.22) a solution exp(ν) =exp(−λ) = (a− cr)2/a2. Now go back to the Einstein equations Gν

µ = −8πT νµ ; if

one thinks of β as creating matter we can then calculate the matter density andpressure. From (1.22) the density is given by 8πρ = −(3c2/a2) + (4c/ar) and theradial pressure is Pr = −ρ (so there is tension rather than pressure). One notesthat Pr = −ρ has been used as the equation of state of prematter in cosmology (cf.[502]). Finally one can calculate the transverse pressure from (1.22) as 8πPt =(3c2/a2) − (2c/ar) (which is anisotropic). Now suppose there is a sphericallysymmetric body filled with matter described by 8πρ = −3c2/a2 +4c/ar, Pr = −ρ,and 8πPt = (3c2/a2(−(2c/ar). Since the matter density can take only nonnegativevalues one has a limit on the size of the body rboundary ≤ (4a/3c). Several modelsare possible; take e.g. a body with maximum radius rbound = 4a/3c. One seesthat on the boundary the density and radial pressure vanish so that this is an openmodel. Go back for a moment to the first equation in (1.3). It may be integrated,giving exp(−λ) = exp(ν) = 1− (8π/r)

∫ r

0ρr2dr. Assume that outside of the body

the Einstein gauge holds, i.e. β = 1 (b = 0) for r > (4a/3c) so that one is left withthe ordinary Riemannian geometry and with the exterior Schwarzschld solutionexp(−λ) = exp(ν) = 1− (2m/r). Comparing and using the equations at hand oneobtains

(1.23) m = 4π∫ 4a/3c

0

ρr2dr = (16a/27c) = (4/9)rbound

Note that in the body (at rs = a/c) there is a singularity of β and of the metric;however the physical quantities ρ, Pr, Pt are regular there (cf. [503]). An externalobserver staying in the Riemannian spacetime will recognize the above entity, madeof Weyl-Dirac geometry, as a body having mass (1.23) and radius 4a/3c.

EXAMPLE 1.2. Another example of matter creation via geometry is alsogiven in [499] with a homogeneous and isotropic FRW universe and line element

(1.24) ds2 = dt2 −R2(t)[

dr2

1− kr2+ r2dΩ2

]Here R(t) is the cosmic scale factor, k = 0, ±1 stands for the spatial curvatureparameter, and dΩ2 = dθ2 + Sin2(θ)dφ2 is the line element on the unit sphere.The universe is filled with ordinary cosmic matter in the state of a perfect fluidat rest and with the cosmic scalar fields β and w. One considers, for κ ≥ 0,T 0

0 = ρ(t), T 11 = T 2

2 = T 33 = −P (t), T = ρ − 3P, β = β(t), and = w(t). Use

also the Einstein gauge β = 1 ∼ b = 0 and the Einstein-Friedmann equations (cf.[499]). After much calculation one looks at the expansion of the universe (withno ordinary matter) and the model provides a high rate of matter creation froman initial empty egg (i.e. geometry brings matter into being). Another modelalong similar lines looks at interaction between geometric fields and matter duringradiation and dust dominated periods with a number of interesting results. Inparticular matter creation takes place in the radiation dominated universe and


also for open and flat models in a dust dominated era while in a closed dustuniverse there is matter creation for awhile after which matter annihilation arisesstimulated by the w field.

EXAMPLE 1.3. We go now to the second paper in [498] which buildsup a singularity free cosmological model that originates from pure geometry.The Planckian state (characterized by ρP = c3/G = 3.83 · 1065 cm−2, RI =(3/8πρP )1/2 = 5.58 · 10−34 cm, and TI = 2.65 · 10−180 K) is preceeded by a pre-Planckian period. This starts from a primary empty spacetime entity, describedby an integrable WD geometry. During the pre-Planckian period geometry createscosmic matter and a the end of this creation process one has the Planckian cosmicegg filled with prematter. The prematter model of [499] will be updated accordingto present observational data and also modified by the introduction of a nonzerocosmological constant. Thus, reviewing a bit, we have gµν , β, and wµ as abovewith wµ = ∂−µw to provide an integrable WD theory. There is an action (1.6) andone uses (1.7) and Wµ = wµ+bµ. Also k−6 = 16πκ (here σ is used in place of κ andlater σ becomes −κ2 so we will think of κ → −κ2 later on. As before one has (1.8),16πS = δLM/δw, (1.9), (1.10), (1.11), 16πB = δLM/δB, IM =

∫LM

√−gd4x,(1.14), S + T = βB, and Tλ

µ;λ − Tbµ = SWµ. Recall also (1.11) is a corollary sothat β and the Dirac charge B can be chosen arbitrarily. Further (1.15), (1.16),etc. will still apply after which one has an Int-W-D theory. One considers a ho-mogeneous isotropic spatially closed (k = 1) universe described by the FRW lineelement of (1.24). For a universe filled with cosmic matter in the state of a perfectfluid at rest its EM tensor has nonvanishing components T i

i (i = 0, 1, 2, 3) andin addition to any matter field there are two cosmic scalar fields β(t) and W (t)stemming from the geometric framework. Taking into account (1.24), one obtainsfrom (1.9) the cosmological equations

(1.25)R2

R2=

8πT 00

3β2− 8πσW 2

3− 2Rb

R− b2 +

β2Λ3− 1

R2

R

R=

4π

β2

(T 1

1 −T 0

0

3

)+

16πσW 2

3− b− Rb

R+

β2Λ3

From (1.9) one also gets the trace equation

(1.26) Gσσ = −8πT

β2− 16πσW 2 + 6b + 18

Rb

R+ 6b2 − 4βΛ

Comparing (1.25) with the usual Einstein equations for cosmology one concludesthat the observable density and pressure of the cosmic matter are ρ = T 0

0 /β2 andP = −T 1

1 /β2. Now we let σ → −κ2 and regard β(t) as a function of R(t) so that

(1.27) β = β(R), β = β′R; β = β′′R2 + β′R

Taking into account (1.27) one can rewrite (1.25) as

(1.28)R2

R2

(1 +

β′Rβ

)2

=8π

3(ρ + κ2W 2) +

β2Λ3− 1

R2;

R

R

(1 +

β′Rβ

)+

R2

R2

(β′′R2

β− (β′)2R2

β2+

β′Rβ

)= −4π

3(3P + ρ + 4κ2W 2) +

β2Λ3


Further the equation (1.8) of the W field takes the form

(1.29) W +

(2β

β+

3R

R

)= − S

2κ2β2≡ ∂t(Wβ2R3) = −SR3

2κ2

With (1.24) (and the density and pressure terms above) the EM relation for matteris now

(1.30) ρ + 3(R/R)(ρ + P ) + (β/β)(ρ + 3P ) = (SW/β2)

and S + T = βB can be written as

(1.31) S + (ρ− 3P )β2 −Bβ = 0

For the FRW line element (1.24) the β field equation (1.11) takes the form (σ ∼−κ2)

(1.32) Rσσ + κ

[(bR3),t

R3+ b2

]+ 16πσ

[(wR3),t

R3− w2

]− 4β2Λ− 8πβ−1B = 0

By (1.29) and (1.31) this turns out to be identical with the trace equation (1.26)and B may be cancelled from the equations (i.e. β and B are arbitrary - recallthat (1.11) is a corollary, etc.). Now introduce the energy density and pressure ofthe W field via

(1.33) ρw = Θ00; Pw = −Θ1

1 = −Θ22 = −Θ3

3

Making use of (1.24) one obtains ρw = Pw = κ2β2W 2 leading to

(1.34) (ρ + κ2W 2),t +β

β(ρ + 3P + 4κ2W 2) +

3R

R(ρ + P + 2κ2W 2) = 0

Then introduce the reduced energy density and pressure ρw and Pw via

(1.35) ρw = ρw/β2; Pw = Pw/β2; ρw = Pw = κ2W 2

and write ρ = ρ + ρw = ρ + κ2W 2 and P = P + Pw = P + κ2W 2; then (1.28)becomes

(1.36)R2

R2

(1 +

β′Rβ

)2

=8πρ

3+

β2Λ3− 1

R2;

R

R

(1 +

β′Rβ

)+

R2

R2

(β′′R2

β− (β′)2R2

β2+

β′Rβ

)= −4π

3(3P + ρ) +

β2Λ3

and the energy momentum relation (1.34) is

(1.37) ρ +β

β(ρ + 3P ) + 3

R

R(ρ + P ) = 0

Now for the pre-Planckian period one looks for matter production by geometryand returns to (1.29) and (1.30). From (1.30) one sees that the W field can act asa creator of matter even if at the beginning moment no matter was present (surelysomeone has thought of a Higgs role for the W field ?). According to (1.29) the Wfield depends on the source function S. On the whole one adopts the singularityfree cosmological model of [502] with its initial prematter period but completedwith a nonzero cosmological constant. Also some constants such as the Hubble

constant, matter densities, etc. are updated. The initial Planckian egg is thuspreceded by a pre-Planckian period originating from a primary geometric state(primary state) at R0 = 5.58 ·10−36 cm and lasting up to the initial Planckian eggat RI = 5.58 · 10−34 cm. This spherically symmetric homogeneous and isotropicuniverse is described by the Int-W-D geometry (no cosmic matter) with nothingbut geometry, including the W and β fields, at R0 = 5.58 ·10−36 cm, ρ0 = 0, P0 =0, β0 = 0, and W0 = 0. Assume this was quasistatic with R0 = 0 and via (1.27)also β0 = 0. Thus one obtains from (1.28) at the beginning moment

(1.38) (8π/3)κ2W 20 + (1/3)β2

0Λ− (1/R20) = 0

Since B may be chosen arbitrarily one can take a suitable function for the Weylianscalar charge S and then calculate the Dirac charge B according to (1.31). An“appropriate” choice is S = S0(β2β/R3 with S0 constant (explanation omitted).Inserting this into (1.29) and integrating one gets W = −(S0/6κ2)(β/R3) so W = 0and one can rewrite (1.38) as

(1.39) (8π/3)(S20/36κ2)(β2

0/R60) + (1/3)Λβ2

0 − (1/R20) = 0

One will use this below to calculate the value of β0 but first the scenario of the veryearly universe must be completed by an equation of state of cosmic matter duringthe pre-Planckian period. According to (1.30) and (1.34) the matter is created bythe W field which has an EM density tensor (1.14), etc. The components of thistensor are related by (1.33), etc., so that the pressure of this field is equal to itsenergy density. Thus one writes P = ρ and then one can rewrite (1.30) as

(1.40) ρ + 6(R/R)ρ + 4(β/β)ρ = SW/β2 ≡ ρ = (1/β4R6)∫

SWβ2R6dt

The density of matter created by geometry is given now by the expression

(1.41) ρ = (S20β6

0/36κ2β4R6)[1− (β/β0)6]

Thus there was a zero matter density and pressure (with the assumptions above) atthe beginning moment and for a rapidly decreasing β(R) one can have (βI/β0)6 <<1 where βI = β(RI). According to this scenario at R = RI the matter densityreaches its maximum ρP = 3.83 · 1065 cm−2 there so that from (1.41), etc., onegets ρP = S2

0β60/36κ2β4

I R6I . From this one can calculate S2

0/36κ2 for a given gaugefunction β(R). Now for a moment go back to the energy equation (1.40); it canbe rewritten as

(1.42) ρ + 6(R/R)ρ + 4(β/β)ρ = −(S20/6κ2)(β2/R6)(β/β)

Then the term on the right side of (1.42), which describes matter creation by theW field can be compared with the third term on the left side which represents theexisting amount of matter. Making use of (1.41) this gives

(1.43) (1/4ρ)(S20β2/6κ2R6) = [3β6/2(β6

0 − β6)]

Further, comparing the matter energy with that of the W field in the equations(1.28) one gets

(1.44) (κ2W 2/ρ) = [β6/(β60 − β6)]

2. REMARKS ON COSMOLOGY 151

Thus in the beginning when β0−β is small W dominates the matter creation whilefor large R, when β << β0, the matter creation term becomes negligible.

This is just a sampling of results in [498, 499, 500, 501, 502, 503]. Manyother cosmological questions of great interest including dark matter, quintessence,etc. are also treated. There are in addition many fascinating papers speculatingabout the original universe from many points of view and we attempt no surveyhere. The approach here via Weyl-Dirac geometry seems however too lovely toignore and it may provide further insight into questions of quantum fluctuations.The inroads into cosmology here are an inevitable consequence of the presence ofWeyl-Dirac theory in dealing with quantum fluctuations and once wave functinsand Bohmian ideas are introduced the quantum potential will automatically arisevia β and wµ.

2. REMARKS ON COSMOLOGY

We begin with some background information (cf. [1, 186, 187, 188, 189,741, 882, 883, 903]. Thus recall that the deBroglie wave length is λ = /p andthe Compton wave length is Λ = /mc. The uncertainty principle states that(∆x)(∆p) ≥ and the diffusion coefficient for Brownian motion is proportional toD = /m. The fractal dimension of a quantum path is df = 2 at scales between λand Λ but becomes df = 1 at scales smaller than Λ. Brownian motion characterizesthe domain between Λ and λ (cf. [1, 903]). Heuristically from ∆p = m(∆x/∆t)and ∆x∆p ≥ we have m(∆x/∆t) = ∆p ≥ /∆x or (∆x)2 ≥ (/m)∆t which canbe rephrased as < x2 >∼ Dt for D = /m. Further note that from E ∼ p2/2mone has ∆E ∼ (1/2m)(∆p)2 (working around p0 = 0 say). Then from ∆p ≥ /∆xand (∆x)2 ∼ D∆t (with ∆x∆p ≥ ) we obtain (1/2m)(∆p)2 ≥ (2/2mD∆t) fromwhich ∆E∆t ∼ /2.

Now from [883] one looks at Weyl geometry and refers to the Lagrangian of[840] of the form L = LC(q, q, t) + γ(2/m)R(q, t) where R is the Ricci scalarcurvature in the Weyl geometry (cf. Section 3.3). Then it turns out that thequantum potential Q has the form Q = −γ(2/m)R and the Q can be related toquantum fluctuations via Fisher information. In [883] one replaces the Weyl vectorφ (which measures length dilations) by a noncommutative (NC) geometry ds2 =(hµν + hµν)dxµdxν with a tensor density hµν arising via the antisymmetric part.This corresponds then to [dxµ, dxν ] ∼ 2 = 0 and in a certain sense legitimizesthe approach of [840]. Moreover the NC geometry produces a multiply connectedspace in which a closed circuit cannot be shrunk to a point so for a circle C ofdiameter λ in e.g. a doubly connected space one will have (V = (/m)∇S and vis some average velocity)

(2.1) Γ =∫

C

mV · dr =

∫C

∇S · dr =

∮dS = mvπλ = π

Consequently λ = /mv and this shows an emergence of the deBroglie wavelengthfollowing from the NC geometry. We note also from [188, 189, 840] that forψ =

√ρexp(iS/) the Weyl vector φ ∼ −∇log(ρ) and ψ satisfies (for Ak = 0) an


equation

(2.2) iψt = − 2

2m

(1√

g∂i√

g

)gik∂kψ +

(V − γ2

mR

)ψ

where R is the Riemannian curvature. Here grad(f) ∼ ∂kf and ∆f correspondsto taking the divergence of the associated contravariant vector gik∂kf , i.e. ∆f =div(grad(f)) = (1/

√g)∂i

√ggik∂kf). Note also in forming a Lagrangian xα =

2gαβpβ or pα = (1/2)gαβ xβ so that (cf. [12])

(2.3) H = gαβpαpβ → L = xαpα − gαβpαpβ = (1/4)gαβ xαxβ

Thus one has a complete geometrization of quantum mechanics (QM) via (2.2)(recall also that the Ricci-Weyl curvature has the form

(2.4) R = R + (1/2γ√

ρ)[(1/√

g)∂i(√

ggik∂k√

ρ)]

where γ = 1/12 here.

REMARK 4.2.1. We recall now that in the Nottale derivation of theSchrodinger equation SE) one has a complex velocity V − iU due to fractalquantum paths (cf. [186, 187, 188, 272, 715]). Here U = D(d/dx)(log(ρ))can be “conveniently” taken to be a constant α (cf. [272]) which would implylog(ρ) = (α/D)x = βx and Q = −(2/8m)β2 as pointed out in [882] (recallQ = −(2/2m)(∂2√ρ/

√ρ) with

√ρ = exp(βx/2) here). Now one can apparently

make a case for the Zitterbewegung or self interaction effects within a minimumcutoff Compton wavelength to generate inertial mass. If Q is inertial energy, sayQ = −δmc2, with (2/8m)β2 = (2/8m)(α2/D2) = mα2/8 = δmc2 one arrives atα ∼ (8δ)c (omitting constants such as 8δ etc. in approximations involving largeand small numbers). It is then argued that the stochastic-fractal formulation ofNottale leads to the emergence of spacetime coordinates (x, ict) and such mattersare obviously intriguing.

There is a great deal of fascinating information available concerning vari-ous fundamental constants and large numbers in physics (we refer here to [604,715, 723, 961, 972] for example and for more “adventurous” material to e.g.[850, 837, 884, 885, 886, 887, 888] and the numerous papers of M. El Naschiein the journal Chaos, Solitons, and Fractals). Dirac had previously spoken elo-quently about the importance of large numbers and their relations and in this spiritone is compelled to look as such matters. One seems at times to be simply playingwith numbers (sometimes called numerology) but there are too many remarkablecoincidences to be ignored and e.g. El Naschie’s program of tying matters togethervia Cantor sets and the golden mean φ is in my opinion worth serious considera-tion. Sidharth’s arguments about Cantorian E∞ also lend some structural meaningand should be pursued further. In any event we gather here first a collection ofnumbers and ideas in no particular order following [882, 883, 884, 885].

(1) The average distance covered in N steps in a random walk is = R/√

Nwhere R is the dimension of the system. Such a relation with R ∼ 1028

cm (radius of the universe) and N ∼ 1080 (number of particles in theuniverse) gives ∼ 10−12 ∼ 10−13 = π which is the Compton wavelength


of a typical elementary particle (the pion). The stipulation that 10−12 ∼10−13 is of course reasonable but somewhat unsettling).

(2) The Planck scale is defined via P = (G/c3)1/2 ∼ 10−33 cm and tP =(G/c5)1/2 ∼ 10−42 sec with mP = 10−5 gm the Planck mass. Here tPis also the Compton time for the Planck mass (P = ctP ).

(3) R ∼ cT where T is the age of the universe so, from item 1, T ∼√

Nτwhere τ = π/c is the Compton time for a pion. Further M = Nm whereM is the mass of the universe and m is the (typical) pion mass.

(4) The energy of fluctuations of the magnetic field B in a region of length is B2 ∼ c/4 so for ∼ Compton wave length the resultant particle fluc-tuation energy in a volume ∼ 3 is 3B2 ∼ c/ = mc2. Thus the entireenergy of an elementary particle of mass m is generated by fluctuationsalone.

(5) The fluctuation in particle number is of order√

N (cf. [483]) and atypical time interval of uncertainty is ∆t ∼ /mc2 (via ∆E∆t ∼ ). Inthe spirit of Prigogine Heisenberg uncertainty gives rise to productionof energy over short intervals of time leading to a one way creation ofparticles. Thus dN/dt ∼

√N(mc2/) leading to T ∼ (/2mc2)

√N =√

Nτ as in item 3 (τ = /c = (/mc)/c - a factor of 2 is included here).(6) Now recall R ∼ GM/c2 where Nm = M where m is the mass of a

typical elementary particle. Then random walk considerations and fluc-tuations of order

√N from the ZPF give R =

√N. Going to [888] we

note first for H = Gm3c/2 (H ∼ Hubble constant) and R = GmN/c2

one has, for G constant, R = (Gm/c2)N ∼ (Gm/c2)(mc2√

N/) =(Gm3c/2)(/mc)

√N = HR as it should. In fact H is often defined as

R/R. In particular one can conclude from R = RH and H constant thatR = RH2.

Consider now [715, 716, 814] where scale relations in micro and macro physicsabound. Let us begin with H = Gm3c/2 or m = (2H/cG)1/3 (m presumablyrefers to pion mass here). Then one notes that the cosmological constant Λ hasdimension 1/L2 and this suggests that there should be a maximal scale lengthL = 1/

√Λ. Next a version of Mach’s principle is achieved by requiring that the

gravitational energy of interaction of a body with the universe (described as amass M at average distance R) should be equal to its self energy of inertial originE = mc2, namely

(2.5) GmM/R = mc2 ⇒ (GM/Rc2) ∼ 1

Now 2GM/c2 corresponds to the classical radius of a Schwartzschild black hole so(2.5) says that the universe is like a black hole. Next imagine M = 4πρR3/3 withR = c = RH so from 2GM/c2R = 1 there results (2G/c2R)(4πρR3/3) = 1 whichimplies (8πGρ/3H2) = Ω = 1 (space flatness condition - with a cosmologicalconstant the Schwartzschild relation is (2GM/c2R) + (ΛR2/3) = 1 ∼ (8πGρ +Λc2)/3H2 = 1 - cf. [716, 720]). Another formula arises by introducing the Planckmass as a natural unit and writing Newton’s law with Gm2

P = c (following fromR = /mc and (Gm2/R) = mc2 which implies Gm2 = c for m = mP ) in the formF = c[(m/mP )(m′/mP )]/R2 (since cmm′/m2

P R2 = Gmm′/R2); such a formula

appears also in [176, 417]. Regarding the cosmological constant one notes firstthat the Planck length ΛP = (G/c3)1/2 is the only length that can be envisionedwith the three fundamental constants , G, c. Here the maximum scale length Lshould have the form L = ΛP K = 1/

√Λ. This gives a number K ∼ 1061. Now

if the universe is a black hole, looking at a resolution scale 1/L in the Einsteinmodel the maximal separation between points is πL so the effective mass should becharacterized by 2GM/c2πL = 1. This leads to one of the classical large numbercoincidences m/mP = (π/2)K which for K ∼ 1061 gives a characteristic mass∼ 1023 solar masses (which corresponds to 1011 galaxies of 1012 solar masses). Toget m/mP = (π/2)K one uses an argument comparing lengths = Gm/ < v2 >and λ = /mv which if equivalent yields Planck mass with v = c and Gm2

P = cor m2

P = c/G. Then from 2GM/c62πL = 1 one has

(2.6)π

2K =

GM

c2L

L

ΛP=

GM

c2ΛP=

cM

m2P c2ΛP

=M

m2P cΛP

=M

mP

Finally consider a characteristic minimal energy Emin = c/L and, for the electronof purely electromagnetic (EM) origin, a scale r0 is defined where e2/r0 = mec

2 (i.e.r0 = αλC is the classical radius of the electron - note r0 = e2/mc2 = λ(/mc) =αλC where λC is the Compton wavelength and α = e2/c or α = e2/c is the finestructure constan - sometimes written as α = e2/4πc in suitable units). Thenassume the gravitational self energy of the electron at scale r0 equals the minimalenergy Emin; this implies Gm2(r0)/r0 = c/L (here m(r0) ∼ α−1me modulo asmall scale dependence of α) and leads to α(mP /me) = K1/3. To see this writeG = c/m2

P as before and recll m = α−1me; then

(2.7)Gm2

r0=

c

L=

c

ΛP K⇒ cm2

r0m2P

=c

ΛP K⇒ K =

r0m2P

m2ΛP

But ΛP = /mP c and r0 = /mc so K = m2P /m3cΛP = m4

P /m3 which meansK1/3 = mP /m = α(mP /me). For completeness we note

(2.8) P =(

G

c3

)1/2

∼ 1.62 · 10−33 cm; λP =P

c∼ 5.4 · 10−44 sec;

mP =

P c∼ 2.17 · 10−5 gram

We jump ahead now to the more recent articles [716, 720, 717, 814] and tothe discussion in [220, 221, 222] (we also find it curious that Nottale’s scalerelativity has not been “blessed” with any establishment interest). We recallfirst a few basic facts following [814]. The simplest form for a scale differen-tial equation describing the dependence of a fractal coordinate X in terms ofresolution ε is given by a first order, linear, renormalization group like equa-tion (∂X(t, ε)/∂log(ε)) = a − δX with solution X(t, ε) = x(t)[1 + ζ(t)(λ/ε)δ].This involves a fractal asymptotic behavior at small scales with fractal dimen-sion D = 1 + δ, which is broken at large scale beyond the transition scale λ (cf.[715, 716, 720] for more detail). By differentiating one obtains dX = dx + dξwhere dξ is a scale dependent fractal part and dx is a scale independent classical

part such that dx = vdt and dξ = η√

2D(dt2)1/2D where < η >= 0 and < η2 >= 1(one considers here only D = 2). Here each individual trajectory is assumed frac-tal and the test paricles can follow an infinity of possible trajectories. This leadsone to a nondeterministic, fluid like description, in terms of v = v(x(t), t). Thereflection invariance dt → −dt is broken via nondifferentiablity leading to a twovalued velocity vector (cf. [186, 187, 272, 715]) and one arrives at a complextime derivative (d′/dt) = ∂t + V · ∇ − iD∆. Setting ψ = exp(iS/2mD) one ob-tains a geodesic equation in fractal space via (d′/dt)V = 0 which becomes thefree Schrodinger equation (SE) D2∆ψ + iD∂tψ = 0. Now in [814] one is inter-ested in macrophysical applications and considers a free particle in a curved spacetime whose spatial part is also fractal beyond some time or space transition. Theequation of motion can be written (to first order approximation) by a free motiongeodesic equation combining the relativistic covariant derivative (describing curva-ture) and the scale relativistic covariant derivative d′/dt (describing fracticality).We refer to [188, 189, 873] for variations on this. Thus one considers in theNewtonian limit

(2.9) (D/dt)V = (d′/dt)V +∇(φ/m) = 0

where φ is the Newton potential energy. In terms of ψ one obtains

(2.10) D2∆ψ + iD∂tψ = (φ/2m)ψ

Since the imaginary part of this equation is the equation of continuity (and think-ing of the motion in terms of an infinite family of geodesics) ρ = ψψ† can beinterpreted as the probability density of the particle postions. For a Kepler po-tential (in the stationary case) one has then

(2.11) 2D2∆ψ + [(E/m) + (GM/r)]ψ = 0

Via the equivalence principle (cf. [13]) this must be independent of the test particlemass while GM provides the natural length unit; hence D = (GM/2w) where wis a fundamental constant with the dimensions of velocity. The ratio αg = w/cactually plays the role of a macroscopic gravitation coupling constant (cf. [13]).One shows in [814] that the solutions of this gravitational SE are characterizedby a universal quantization of velocities in terms of the constant w = 144.7± 0.5km/s (or its multiples or submultiples); the precise law of quantization dependson the potential. Depending on the scale either the classical or the fractal partdominates. Various situations are examined and we only indicate a few here.The evolution equations are the Schrodinger - Newton equation and the classicalPoisson equation (cf. [632])

(2.12) D2∆ψ + iD∂ψ

∂t− φ

2mψ = 0; ∆Φ = 4πGρ (φ = mΦ)

Here Φ is the potential and φ = mΦ the potential energy. Separating the real andimaginary parts one arives at

(2.13)

m(

∂∂t + V · ∇

)V = −∇(φ + Q); ∂P

∂t + div(PV ) = 0; Q = −2mD2 ∆√

P√P


In the situation where the particles are assumed to fill the “orbitals” the density ofmatter becomes proportional to the probability density, i.e. ρ ∝ P = ψψ† and thetwo equations combine to form a single Hartree equation for matter alone, namely

(2.14) ∆(D2∆ψ + iD∂tψ

ψ

)− 2πGρ0|ψ|2 = 0

Another case arises when the number of bodies is small and they follow at ran-dom one among the possible trajectories so that P = ψψ† is nothing else than aprobability density while space remains essentially empty. It is suggested that thisallows one to explain some effects that up to now have been attributed to darkmatter (cf. [290, 716, 720] and below for more on this).

We mention in particular the case where Φ = −GM/r and the SE becomesD2∆ψ+iD∂tψ+(GM/2r)ψ = 0. One looks for solutions ψ = ψ(r)exp(−iEt/2mD)and makes substitutions /2m ∼ D and e2 ∼ GMm where m is the test parti-cle inertial mass. This yields an equation similar to the quantum hydrogen atomequation whose solution involves Laguerre polynomials ψ(r) = ψnm(r, θ, φ) =Rn(r)Y m

(θ, φ) and the energy/mass ratio is quantized as

(2.15) En/m = −(G2M2/8D2n2) = −(1/2)(w20/n2)

while the natural length unit is the Bohr radius a0 = 4D2/GM = GM/w20. Con-

sider now particles such as gas, dust, etc. in a highly chaotic and irreversiblemotion in a central Kepler potential; via the SE (2.11) there are solutions char-acterized by well defined and quantized values of conservative quantities such asenergy etc. One therefore expects the particles to self-organize into “orbitals” andthen to form planets etc. by accretion. Once so accreted one can recover classicalelements such as eccentricity, semi-major axis, etc. We refer to [716, 720, 814]for more discussion and details of this and many other examples.

One refers later in the paper [814] to the dark matter problem and its connec-tion to the formation of galaxies and large scale structure. Scale relativity allowsone to suggest some solutions. Indeed the fractal geometry of a nondifferentiablespace time solves the problem of formation on many scales and it also impliesthe appearance of a new scalar potential (as in (2.13) which manifests the frac-tality of space in the same way as Newton’s potential manifests its curvature. Itis suggested that this new potential (Q) may explain the anomalous dynamicaleffects without needing any missing mass. In this direction consider the case ofthe flat rotation curves of spiral galaxies. The formation of an isolated galaxyfrom a cosmological background of uniform density is obtaind in its first steps asthe fundamental level solution n = 0 of the SE with an harmonic oscillator grav-itational potential (2π/3)Gρr2 (for which some details are worked out in [814]).Once the galaxy is formed let r0 be its outer radius beyond which the amountof visible matter becomes small. The potential energy at this point is given viaφ0 = −(GMm/r0) = −mv2

0 where M ∼ mass of the galaxy. Observational datasays that the velocity in the exterior region of the galaxy keeps the constant valuev0 and from the virial theorem the potential energy is proportional to the kineticenergy so that it also keeps the constant value φ0 = −GMm/r0. Therefore r0

3. WDW EQUATION 157

is the distance at which the rotation curve begins to be flat and v0 is the cor-responding constant velocity. In the standard approach this flat rotation curveis in contradiction with the visible matter alone from which one would expect toobserve a variable Keplerian potential energy φ = −GMm/r. This means thatone observes an additional potential energy Qobs = −(GMm/r0)[1− (r0/r)]. Nowthe regions exterior to the galaxy are described in the scale relativity approach bya SE with a Kepler potential enery φ = −GMm/r where M is still the sole visiblemass, since we assume here no dark matter. The radial solution for the funda-mental level is

√P = 2exp(−r/rB) where rB = GM/w2

0 is the macroscopic Bohrradius of the galaxy. Now one computes the theoretically predicted new potentialQ from (2.13) (using D = GM/2w0) to get

(2.16) Qpred = −2mD2 ∆√

P√P

= −GMm

2rB

(1− 2rB

r

)= −1

2w2

0

(1− 2rB

r

)Thus one obtains, without any added hypotheses, the observed form Qobs of thenew potential. Moreover the visible radius and the Bohr radius are now relatedvia r0 = 2rB . The constant velocity v0 of the flat rotation curve is also linkedto the fundamental gravitational constant w0 via w0 =

√2v0. Observational data

supporting all this is also given.

3. WDW EQUATION

We go now to the famous Wheeler-deWitt equation (which might also bethought of as an Einstein-Schrodinger equation). Some background about this isgiven in Appendix C along with an introduction to the Ashtekar variables (fol-lowing the beautiful exposition of [69]). The approach here follows [870, 876]which provides a Bohmian interpretation of quantum gravity and we cite also[55, 56, 57, 58, 62, 70, 119, 303, 394, 476, 545, 551, 556, 630, 665, 819,820, 896, 897, 929, 930, 931, 932] for material on WDW and quantum gravity.Extracting liberally (and optimistically) now from [876] (first paper) one writesthe Lagrangian density for general relativity (GR) in the form (16πG = 1)

(3.1) L =√−gR =

√qN(3R + Tr(K2))

where 3R is the 3-dimensional Ricci scalar, Kij is the extrinsic curvature, and qij

is the induced spatial metric. The canonical momentum of the 3-metric is givenby

(3.2) pij =∂L

∂qij=√

q(Kij − qijTr(K))

The classial Hamiltonian is

(3.3) H =∫

d3xH; H =√

q(NC + N iCi)

where the lapse and shift functions, N and Ni, are given via (cf. [69])

(3.4) C = 3R +

1q

(Tr(p2)− 1

2(Tr(p))2

)= −2Gµνnµnν ;


Ci = −23∇j

(pij√

q

)= −2Gµin

µ

Here nµ is the normal vector to the spatial hypersurfaces given by nµ = (1/N,− N/N).Now in the Bohmian approach one must add the quantum potential to the Hamil-tonian to get the correct equations of motion so H → H + Q via H → Q where

(3.5) Q =∫

d3xQ; Q = 2NqGijk1|ψ|

δ2|ψ|δqijδk

Here Gijk is the superspace metric and ψ is the wavefunction satisfying the WDWequation. This means that we must modify the classical constraints via

(3.6) C → C +Q√

qN; Ci → Ci

Now for the constraint algebra one uses the integrated forms of the constraintsdefined as

(3.7) C(N) =∫

d3x√

qNC; C( N)∫

d3x√

qN iCi

Then (cf. [69, 870] and Appendix C for notation)

(3.8) C( N), C( N ′) = C( N · ∇ N ′ − N ′ · ∇ N) ≡ C(N i∇N ′i −N

′i∇Ni);

C( N), C(N) = C( N · ∇N); C(N), C(N ′) ∼ 0The first 3-diffeomorphism subalgebra does not change with respect to the classicalsituation and the second, representing the fact that the Hamiltonian constraint isa scalar under the 3-diffeomorphism, is also the same as in the classical case.In the third the quantum potential changes the Hamiltonian constraint algebradramatically giving a result weakly equal to zero (i.e. zero when the equationsof motion are satisfied). Following [876] we will give a number of calculationsnow regarding this Hamiltonian constraint. Thus first write the Poisson bracketexplicitly as

(3.9) C(N), C(N ′) =∫

d3z√

q(z)(

δC(N)δqij(z)

δC(N ′)δpij(z)

− δC(N)δpij(z)

δC(N ′)δqij(z)

)=

= C(N ∇N ′ −N ′∇N) + 2∫

d3zd3x√

q(z)Gijk(z)pk(z)×

×(−N(z)N ′(x) + N(x)N ′(z))δ(Q/

√qN))

δqij(z)To simplify one differentiates the Bohmian HJ equation to get (cf. [870])

(3.10)1N

δ

δqij

Q√

q=

34√

qqkp

ijpkδ(x− z)−√

q

2qij(3R− 2Λ)δ(x− z)−√q

δ3R

δqij

and use this in evaluation of the Poisson bracket giving the result indicated in(3.8). This calculation is given in the Appendix to [876] and we repeat it here forclarity.

REMARK 4.3.1. We follow here the Appendix to [876] and to evaluate theintegral (3.9), in view of (3.10), one needs to consider

∫d3zF (q, p,N,N ′)(δ 3R/δqij).

3. WDW EQUATION 159

First look at the variation of the Ricci scalar with respect to the metric. Usingthe Palatini identity and dropping the superscript 3 one has

(3.11) δRij = (1/2)qk(∇j∇iδqk −∇k∇jδqi −∇k∇iδqj +∇k∇qij)

Consequently

(3.12)δRij(x)δqab(z)

=12√

qqk

(δakδb

∇j∇iδ(x− z)√

q− δa

δbi∇k∇j

δ(x− z)√

q−

−δa δb

j∇k∇iδ(x− z)√

q+ δa

i δbj∇k∇

δ(x− z)√

q

)and therefore

(3.13)δR(x)δqab

=δ(qijRij)(x)

δqab= −R

abδ(x− z) +√

q(qab∇2 −∇a∇b)δ(x− z)√

q

Using this identity in the equation of motion (3.10) the only nonvanishing termsin (3.9) are(3.14)

C(N), C(N ′) ∼ C(N ∇N ′ −N ′∇N)− 2∫

d3zd3x√

q(x)q(z)Gijk(z)pk(z)

×(N(x)N ′(z)−N(z)N ′(x))(

qij(x)∇2x

δ(x− z)√

q−∇i

x∇jx

δ(x− z)√

q

)where ∼ means the equality is weak (i.e. modulo the equation of motion). Inte-grating by parts gives

(3.15) C(N), C(N ′) ∼ 2∫

d3x(∇j(N∇iN′)−∇j(N ′∇iN))pij+

+∫

d3x√

qGijkpk(N ′∇i∇jN −N∇i∇jN ′)−

−∫

d3x√

qGijkpkqij(N ′∇2N −N∇2N ′)

Hence there results

(3.16) C(N), C(N ′) ∼ 2∫

d3x(N∇j∇iN′ −N ′∇j∇iN)pij+

+∫

d3xpk(N ′∇k∇N −N∇k∇N′ + N ′∇∇kN −N∇∇kN ′−

−qk(N ′∇2N −N∇2N ′)) +∫

d3xqkpk(N ′∇2N −N∇2N ′) = 0

as desired.

One sees that the presence of the quantum potential means that the quantumalgebra is the 3-diffeomorphism algebra times an Abelian subalgebra and the onlydifference with [631] is that this algebra is weakly closed. One sees that the algebra(3.8) is a clear projection of the general coordinate transformations to the spatialand temporal diffeomorphisms and in fact the equations of motion are invariantunder such transformations (cf. also [769]). In particular although the form ofthe quantum potential will depend on regularization and ordering, in the quantum


constraint algebra the form of the quantum potential is not important; the algebraholds independently of the form of the quantum potential. Further it appears thatthe inclusion of matter terms will not change anything.

One goes next to the quantum Einstein equations (QEI). For the dynamicalpart consider the Hamiltonian equations

(3.17) qij = H, qij; pij = H, pijwhich produce the quantum equations (note the square bracket [ ] means thatone is to antisymmetrize over all permutations of the enclosed indices, multiplyingeach term in the sum by the sign (±1) of the permutation)

(3.18) qij =2√

qN

(pij −

12pk

kqij

)+ 2 3∇[iNj]

(3.19) pij = −N√

q

(3R

ij − 12

3Rqij

)+

N

2√

qqij

(pabpab −

12(pa

a)2)−

−2N√

q

(piapj

a −12pa

apij

)+√

q(∇i∇jN − qij 3∇a 3∇aN)+

+√

q 3∇a

(Na

√qpij

)− 2pa[i 3∇aN j] −√q

δQ

δqij

Combining these two equations one obtains after some calculation

(3.20) Gij = − 1

N

δQ

δqij

which means that the quantum force modifies the dynamical part of the Einsteinequations. For the nondynamical parts one uses the constraint relations (3.4) toget

(3.21) G00 =

Q

2N3√q; G

0i = − Q

2N3√qN i

These last two equations can be written via

(3.22) G0µ =

Q

2√−g

g0µ

and the nondynamical parts are also modified by the quantum potential.

One addresses next the possibility that for a reparametrization invariant the-ory the equations obtained by the Hamiltonian may differ from those given by thephase of the wavefunction and the guidance formula (in a Bohmian spirit). How-ever it is seen that there is no difference. Indeed write the Bohmian HJ equation(cf. [123, 477, 557]) by decomposing the phase part of the WDW equation; thisgives

(3.23) GijkδS

δqij

δS

δqk−√q (3R−Q) = 0

where S is the phase of the WDW wave function. In order to get the equationof motion one differentiates the HJ equation with respect to qab and uses the

3. WDW EQUATION 161

guidance formula pk ≡ √q(Kk−qkK) = δS/δqk. After considerable calculationone arrives again at (3.20). Thus the evolution generated by the Hamiltonian iscompatible with the guidance formula, i.e. the Poisson brackets of the Hamiltonianand the guidance relation (χk = pk − δS/δqk) are zero. This can be evaluatedexplicitly and equals zero weakly so consistency prevails.

Next one shows explicitly that these modified Einstein equations (MEI) arecovariant under spatial and temporal diffeomorphisms. Consider first t → t′ = f(t)with x unchanged; one has

(3.24) q′ij = qij ; N ′i = (df/dt)Ni; N ′ = (df/dt)N

Putting these in the MEI one sees that the right side transforms as a second ranktensor under time reparametrization. Similarly consider x → x′ = g(x) with tunchanged; one has

(3.25) q′ij =∂x

∂x′i∂xm

∂x′j qm; N ′i =

∂x

∂x′i N; N ′ = N

Again the right side of MEI is a second rank tensor under a spatial 3-diffeomorphism.Inclusion of matter field is straightforward via

(3.26) Gij = −κT

ij − 1N

δ(QG + Qm)δgij

; G0µ = −κT

0µ +QG + Qm

2√−g

g0µ

where

(3.27) Qm = 2 N√

q

21|ψ|

δ2|ψ|δφ2

where φ is the matter field and, as before,

(3.28) QG = 2NqGijk1|ψ|

δ2|ψ|δqijδqk

(3.29) QG =∫

d3xQG; Qm =∫

d3xQm

Equations (3.26) are the Bohm-Einstein equations which are in fact the quantumversion of the Einstein equations; regularization only affects the quantum poten-tial but the QEI are the same. They are invariant under temporal and spatialdiffeomorphisms and can be written as

(3.30) Gµν = −κT

µν + Sµν

(3.31) S0µ =

QG + Qm

2√−g

g0µ =Q

2√−g

g0µ; Sij = − 1

N

δ(QG + Qm)δgij

= − 1N

δQ

δgij

(Sµ,ν is the quantum correction tensor - under the temporal ⊗ spatial diffeomor-phism subgroup which is peculiar to the ADM decomposition). Note that theQEI were derived for a Robertson-Walker metric in [961] but without symmetryconsiderations or more general metrics. One concludes with the conservation lawvia taking the divergence of (3.30) to get

(3.32) ∇µTµν =

1κ∇µS

µν


REMARK 4.3.2. We refer to [724] for a discussion of the Lichnerowicz-Yorkequation as a solution to the constraint equations of GR.

3.1. CONSTRAINTS IN ASHTEKAR VARIABLES. We go now tothe third paper in [876] where the new variables (or Ashtekar variables) are em-ployed and it is shown that the Poisson bracket of the Hamiltonian with itselfchanges with respect to its classical counterpart but is still weakly equal to zero(as above in Section 4.3). Caution is advised however since ill defined terms havenot been regularized; for this one needs a background metric and the result mustbe independent of such a metric. Thus the dynamical variables are the self dualconnection Ai

a and athe canonical momenta are Eai with constraints given via

(3.33) Gi = DEai ; Cb = Ea

i F iab; H = εij

k Eai Eb

jFkab

where Da represents the self-dual covariant derivative and F kab is the self-dual

curvature (we refer to Appendix for the new variables). Canonical quantization ofthese constraints can be achieved via changing Ea

i → −(δ/δAia) to get

(3.34) Daδψ(A)δAi

a

= 0; F iab

δψ(A)δAi

a

; 2εijk F k

ab

δ2ψ(A)δAi

aδAjb

= 0

In order to get the causal interpretation one puts a definition ψ = Rexp(iS/)into these relations to obtain

(3.35) (A) DaδR(A)δAi

a

= 0; (B) DaδS(A)δAi

a

= 0;

(C) F iab

δR(A)δAi

a

= 0; (D) F iab

δS(A)δAi

a

= 0;

(E) εijk F k

ab

δ

δAia

(R2 δS(A)

δAjb

)= 0; (F) − εij

k F kab

δS(A)δAi

a

δS(A)δAj

b

+ Q = 0

where the quantum potential is defined as

(3.36) Q = −2εijk F k

ab

1R

δ2R(A)δAi

aδAjb

Here (E) is the continuity equation while (F) is the quantum Einstein-Hamilton-Jacobi (EHJ) equation. The quantum trajectories would be achieved via the guid-ance relation

(3.37) Eai = i

δS(A)δAi

a

Now for the constraint algebra; in terms of smeared out Gauss, vector, andscalar constraints (the notation N∼ is not clearly defined here - cf. (3.3))

(3.38) G(Λi) = −i

∫d3xΛiDaEa

i ; H(N∼) =12

∫d3xN∼εij

k Eai Eb

jFkab;

C( N) = i

∫d3xN bEa

i F iab −G(NaAi

a)

the classical algebra is

(3.39) G(Λi),G(Θj) = G(εiijkΛjΘk); C( N), C( M) = C(L M

N);

3. WDW EQUATION 163

C( N),G(Λi) = G(L NΛi); C( N),H(M∼) = H(LN∼M∼)

G(Λi),H(N∼) = 0; H(N∼),H(M∼;) = C( K) + G(KaAia)

where Ka = Eai Ebi(N∼∂bM∼−M∼∂bN∼). The quantum trajectories can now be obtained

from the quantum Hamiltonian given by HQ = H + Q and the smeared outgauge and diffeomorphism constraints will not change; the Hamiltonian constraintbecomes

(3.40) HQ(N∼) =12

∫d3xN∼εij

k Eai Eb

jFkab + Q(N∼)

where Q(N∼) =∫

d3xN∼Q (the notation Q and Q is switched here from previoususe above). The first three constraint Poisson brackets in (3.39) will not changeand the fourth is still valid because the quantum potential is a scalar density andone has

(3.41) C( N),HQ(M∼) = HQ(L NM∼)

This applies also to the fifth bracket because

(3.42) G(Λi),HQ(N∼) = 0

but the last bracket changes via

(3.43) HQ(N∼),HQ(M∼) = H(N∼),H(M∼)+ Q(N∼),H(M∼)+

+H(N∼),Q(M∼)+ Q(N∼),Q(M∼)Here the last term is identically zero, since the quantum potential is a functionalof the connection only. The sum of the second and third terms is

(3.44) Q(N∼),H(M∼)+ H(N∼),Q(M∼) ∼

∼ −∫

d3x(N∼εij

k F kabE

bjDc(M∼εm

i Ea Ec

m)−M∼εijk F k

abEbjDc(N∼εm

i Ea Ec

m))

A calculation then shows that the Poisson bracket of the quantum Hamiltonianwith itself is given via

(3.45) HQ(N∼),HQ(M∼) ∼ 0

which is similar to the situation with the old variables (cf. Remark 4.3.1).

Now in order to obtain the quantum equations of motion via the Hamiltonequations one has

(3.46) Aia = −iεijkN∼Eb

jFabk −N bF iab;

˙E = iεjki Db(N∼Ea

j Ebk);

−2Db(N [aEb]i ) +

i

2

∫d3x

δQ(N∼)δAi

a(z)Further to recover the real quantum general relativity one must set the realityconditions, which are

(3.47) Eai Ebi must be real;

iεijkE(ai Da(Eb)

k Ecj ) +

i

2

∫d3x

δQ

δAi(a(z)

Eb)(x) must be real


(note round brackets ( ) mean symmetrization with respect to the indices con-cerned). Thus formally one can construct a causal version of canonical quantumgravity using the Bohm-deBroglie interpretation of QM. All of the quantum be-havior is encoded in the quantum potential. One has a well defined trajectory andno operators arise; the algebra action is in fact the Poisson bracket and only thePoisson bracket of the Hamiltonian with itself will change relative to the classicalalgebra by being weakly instead of strongly equal to zero. The result is similar tothat obtained above with the old variables and one can give meaning to the ideaof time generator for the Hamiltonian constraint. The equations of motion whenfinally written out should contain the quantum force. Regularization of ill definedterms remains and is promised in forthcoming papers of F. and A. Shojai.

The approach here in Sections 4.3 and 4.3.1 is so clean and beautiful that wesuspend any attempt at criticism. Eventually one will have to reconcile this withresults of [770, 772] for example (cf. Section 4.5). We remark also that in [880]one makes a preliminary study of Bohmian ideas in loop quantum gravity usingAshtekar variables.

4. REMARKS ON REGULARIZATION

In [961] (second paper) for example one considers the classical and WDWdescription of a gravity-minisuperspace model (cf. also [573]). Thus consider ahomogeneous and isotropic metric defined via

(4.1) ds2 = −N(t)2dt2 + a(t)2dΩ23

where dΩ23 is a standard metric on 3-space. The lapse function N and the scale

factor a depend on a time parameter t. A minisuperspace model represented by asingle homogeneous mode φ is defined by the Lagrangian

(4.2) L = −a3

[1

2N

(a

a

)2

+ NVG(a)

]+ a3

[φ2

2N−NVM (φ)

]One uses the Planck mass m2

P = 3/4πG to scale all dimensional quantities soa ≡ amP , φ ≡ φ/mP , etc.; VM is the potential for the scalar mode φ and thegravitational potential VG(a) = −(1/2)Ka−2 +(1/6)Λ may contain a cosmologicalconstant Λ and a curvature constant K = 1, 0, or −1 for a spherical, planar, orhyperspherical 3-space. From the Lagragian one derives now the classical equationsof motion by varying N, a, and φ respectively to obtain

(4.3)12

(a

a

)2

− VG(a) =12φ2 + VM (φ);

12

(a

a

)2

+a

a− 3Vg(a)− a∂aVG(a) + 3

(12φ2 − VM (φ)

)= 0;

φ + 3(

a

a

)φ + ∂φVM (φ) = 0

Here one has chosen the gauge N = 1 and t is identified with the classical time.However the time parameter is not directly observble, only the correlations a(φ)or φ(a) which follow from the solutions of (4.3). One could imagine an additional

4. REMARKS ON REGULARIZATION 165

degree of freedom τ(t) to be used as a clock but this need not be done. Onederives the WDW equation from (4.2) in the standard manner. After computingthe classical Hamiltonian and replacing the canonical momenta pa = −aa andpφ = a3φ by operators pφ → −i∂φ and pa → −i∂a the WDW Hamiltonian is

(4.4) HWDW =[12a−3(a∂a)2 + a3VG(a)

]+[12a−3∂2

φ + a3VM (φ)]

= HG + HM

The WDW equations is HWDW ψ = 0 and there is an operator ordering ambiguity;the “Lagrangian” ordering has been chosen which makes HWDW formaly selfad-joint in the inner product (ψ, ψ) =

∫a2dadφψ∗(a, φ)φ(a, φ). It is not clear that

HWDW can be used as a generator for time evolution since time has disappearedaltogether in (4.4).

Now in the dBB treatment one has

(4.5) S +1

2m(∂xS)2 + V + Q = 0; R2 + ∂x(R2∂xS) = 0

where Q = −∂2xR/2mR. The trajectories are found by solving the autonomous

system x = (1/m)∂xS and the measure R2dx gives the probability for trajectoriescrossing the interval (x, x + dx). Now treat the WDW equation as a SE whichhappens to be time independent and write it in the quantum potential form withψ = R(a, φ)exp(iS(a, φ)) leading to

(4.6)12a−3[−(a∂aS)2 + (∂φS)2] + a3[VG + VM + QGM ] = 0;

−a∂a(R2a∂aS) + ∂φ(R2∂φ) = 0

These equations come from the real and imaginary part of HWDW ψ = 0 and thequantum potential is

(4.7) QGM = −12a−6

[−−(a∂a)2R

R+

∂2φR

R

]Trajectories (a(t), φ(t)) ae obtained from S(a, φ) by identifying ∂aS with the mo-mentum pa and ∂φS with pφ; thus one uses the definition of the canonical momentagiven before (4.4) to define trajectories parametrized by a time parameter t, via

(4.8) a = −a−1∂aS(a, φ); φ = a−3/2∂φS(a, φ)

One notes that the probability measure R2(a, φ)a2dadφ is conserved in time t if(a, φ) are solutions of (4.7). This is a consequence of using the measure a2dadφtogether with the Lagrangian factor ordering making HWDW formally self adjoint.In using t as in (4.8) one should eliminate t after solving in order to determinea(φ) or φ(a); for trajectories where e.g a(t) is 1-1 the scale factor can be used asa clock. The analogues of equations (4.3) are obtained by differentiating the firstequation in (4.6) with respect to a and φ and then eliminating S, using

(4.9) −∂t(aa) = ∂2aSa + ∂φ∂aSφ; ∂t(a3φ) = ∂a∂φa + ∂2

φSφ


leading to

(4.10)12aa2 − VG(a)− a3 1

2φ2 + VM (φ)

]= QGM (a, φ);

12a2 + aa− ∂aVG(a) + a2

[12φ2 − VM (φ)

]= ∂aQGM (a, φ);

φ + 3a

a+ ∂φVM (φ) = −a−3/2∂φQGM (a, φ)

This evidently generalizes (4.3). There is more in this paper which we omit here,namely a semiclassical development is given as well as and some exact solutionsof WDW for toy models.

In [123] one looks at an equation

(4.11) Hψ(x) =(

12gij(x)∇i∇j − V (x)

)ψ(x) =

(12− V (x)

)ψ(x) = 0

Assume ψ = R(x)exp(iS(x)/) with R, S real leading to

(4.12) H[S(x)] =12gij

∂S

∂xi

∂S

∂xj+ V (x) =

2

2RR; RS + 2gij ∂S

∂xi

∂S

∂xj= 0

Introduce time via

(4.13)dxi

dt= gij δH[S(x)]

δ(∂S/∂xj)

This defines the trajectory xi(t) in terms of the phase of the wave function S. Putthis now back into (1.2) to obtain

(4.14)12gij x

ixj + V (x) + Q = 0; Q = − 2

2RR

Now define classical momenta

(4.15) pi =δH[S(x)]δ(∂S/∂xi)

= gij xj

and write (4.14) in the form

(4.16) H =12gijpipj + V (x) + Q = 0; pi = −∂H

∂xi; xi =

∂H

∂pi

This gives a method of identifying the time evolution corresponding to the wavefunction of a WDW type equation (e.g. the wave function of the universe). Thefollowing points are mentioned in [123]:

• (4.16) is equivalent to the classical equations of motion except for thequantum potential which therefore. in some sense, clarifies the effect ofquantum matter on gravity.

• A semiclassical theory can be constructed from the above.

[


• The quantum potential provides a natural way to introduce time evenwhen the Hamiltonian constraint (which doesn’t involve time) is acting.In this connection note that the definition of time is not unique. Onecould use a more general expression

(4.17) xj = N(x)δH[S(x)]δ(∂S/∂xj)

where N could be identified with the lapse function for the ADM formu-lation.

In [123] one also studies situations of the form

(4.18) ds2 = −N2dt2 +3∑1

gii(ωi)2

where N is a lapse function and ωi are appropriate 1-forms (some Bianchi typesare classified) and we refer to [123] for details and [84] for criticism.

In [124] one starts with the classical constraints of Einstein’s gravity, namelythe diffeomorphism and Hamiltonian constraint in the form

(4.19) Da = ∇aπab; H = κ2Gabcdπabπcd − 1

κ2

√h(R + 2Λ)

Here πab are momenta associated with the 3-metric hab where

(4.20) Gabcd =1

2√

h(hachbd + hadhbc − habhcd

is the WDW metric where R is the 3-dimensional scalar curvature, κ is the grav-itational constant, and Λ the cosmological constant. The constraints satisfy thealgebra

(4.21) [D,D] ∼ D; [D,H] ∼ H; [H,H] ∼ D

The rules of quantization given by the metric representation of the canonical com-mutation relations are

(4.22) [πab(x), hcd(y)] = −iδac δb

dδ(x, y); πab(x) = − δ

δhab(x)

There are problems here of all types (cf. [124]) which we will not discuss butin [572] a class of exact solutions of the WDW equation was found via heatkernel regularization of the Hamiltonian with a suitable ordering and the questionaddressed here is the level of arbitrariness in this construction. One bases now suchconstructions on the principle that the algebra of constraints should be anomalyfree, i.e. the algebra should be weakly identical with the classical one. One choosesa starting space of states to consist of integrals over compact 3-space of scalardensities like V =

∫M

√h, R =

∫M

√hR, etc. so ψ = ψ(V,R, · · · ). For the

diffeomorphism constraint one takes the representation Da(x) = −i∇xb (δ/δhab(x))

where ∇xb means the covariant derivative acting at the point x. This constraint


then annihilates all the states and the first commutator relation (1.21) is satisfied.Further the second relation in (4.21) reduces to the formal relation

(4.23) D(Hψ) ∼ Hψ

Now for the construction of the WDW operator one makes a point split in thekinetic term of the form

(4.24) Gabcd(x)πab(x)πcd(x)

∫dx′Kabcd(x, x′, t)

δ

δhab(x)δ

δhcd(x′)

where limt→0+Kabcd(x, x′, t) = δ(x, x′) and in particular one takes

(4.25) Kabcd(x, x′, t) = Gabcd(x′)∆(x, x′, t)(1 + K(x, t));

∆ =exp(−(1/4t)hab(x− x′)a(x− x′)b)

4πt3/2

with K(x, t) analytic in t. Next to resolve the ordering ambiguity in H one adds anew term Lab(x)(δ/δhab(x)) where Lab is a tensor to be derived along with K(x, t).Thus the WDW operator will have the form

(4.26) H(x) = κ2

∫dx′Kabcd(x, x′, t)

δ

δhab(x)δ

δhcd(x′)+

+Lab(x)δ

δhab(x)+

1κ2

√h(R + 2Λ)

Next one needs to define the action of operators on states which involves discussionsof regularization and renormalization. Regularization is a trade off here of + and- powers of t for δ(0) type singularities and for renormalization one drops positivepowers of t and replaces singular terms t−k/2 by renormalization coefficients ρk (cf.[124] for more details and references). Then to interpret (4.23) for example onethinks of an operator acting on a state and the resulting state after renormalizationcan be acted upon by another operator. Thus (4.23) means

(4.27) D(Hψ)ren ∼ (Hψ)ren

Similarly for the Hamiltonian constraint

(4.28) (H[N ](H(M)ψ)ren)ren − (H[M ](H[N ]ψ)ren)ren = 0

(since ψ is diffeomorphism invariant); here one is using the smeared form of theWDW operatr H[M ] =

∫dxM(x)H(x). After some calculation one also concludes

that for the states which are integrals of scalar densities there is no anomaly in thediffeomorphism - Hamiltonian commutator. As for the Hamiltonian - Hamiltoniancommutator one claims that if (Hψ)ren contains terms which contain 4 or morederivatives of the metric like R2, RabR

ab, etc. then (4.28) cannot be satisfied.This was checked for some low order terms but is otherwise open. The form of thewave function must also be examined and this leads to further conditions on theWDW operator, which is finally advanced in the form

(4.29) H(x) = κ2

∫dx′Gagcd(x′)∆(x, x′, t)

δ

δhab(x)δ

δhcd(x′)+

+(

1κ

αhab + κγ

(14habR + Rab

))(x)

δ

δhab(x)+

1κ2

√h(R + 2Λ)


where α, γ are independent constants.

When one now works with a Bohmian approach where ψ = exp(Γ)exp(iΣ)and one considers only the real part of the resulting equation, namely

(4.30) −κ2Gabcd(x)δΣ

δhab(x)δΣ

δhcd(x)+

1κ

√h(x)(R(x)+2Λ)+(L)ab(x)

δΓδhab(x)

−

−(L)ab(x)δΣ

δhab(x)+ e−Γκ2

(δ2eΓ

δh2

)ren

(x) = 0

Identifying now

(4.31) pab(x) =δΣ

δhab(x)

we see that the first two terms in (4.30) are the same as the Hamiltonian constraintof classical relativity. The remaining terms are quantum corrections (and if wereinserted they would all be proportional to 2). The wave function is subject to thesecond set of equations, namely those enforcing the 3-dimensional diffeomorphisminvariance, which read (for the imaginary part)

(4.32) ∇a δΣδhab(x)

= ∇apab = 0

Thus the theory is defined by two equations (1.30) (with the pab inserted) and(4.32). This leads then to the full set of ten equations governing the quantumgravity in the quantum potential approach, namely

(4.33) 0 = Ha = ∇apab;

0 = H⊥ = −κ2Gabcd(x)pabpcd +1κ2

√h(x)(R(x) + 2Λ)+

+(L)ab(x)δΓ

δhab(x)−(L)ab(x)pab + κ2e−Γ

(δ2eΓ

δh2

)ren

(x);

hab(x, t) = hab(x, t),H[N, N ]; pab(x, t) = pab(x, t),H[N, N ]

where (cf. Section 4.3)

(4.34) H[N, N ] =∫

d3x(N(x)H⊥(x) + Na(x)Ha(x))

This all shows in particular that when questions of regularization and renormal-ization are taken into account life becomes more complicated. Various examplesare treated in [123, 124, 571] where the quantum potential approach works verywell but others where time translation becomes a problem.


5. PILOT WAVE COSMOLOGY

We refer here to [74, 75, 77, 84, 85, 86, 81, 769, 770, 771, 772, 881] (otherreferences to be given as we go along). First from [881] a set of nonrelativisticspinless particles are described via spatial coordinates x = (x1, · · · , xn) and thewave function ψ satisfies the SE

(5.1) i∂ψ

∂t= −2

2

∑ 1mn

∆nψ + V ψ

Putting in ψ = Rexp(iS/) gives then

(5.2)∂S

∂t+∑

n

12mn

(∇nS)2 + V + Q = 0; Q = −∑

n

2

2mn

∆nR

R;

∂R2

∂t+∑

n

1mn∇n(R2∇nS) = 0

In the pilot wave interpretation the evolution of coordinates is governed by thephase via mnxn = ∇nS. In the relativistic theory of spin 1/2 one continues todescribe particles by their spatial coordinates but the guidance conditions and theequations for the wave are now different, namely for the multispinor ψα1···αn

(x, t)the Dirac equation is (α ∼ (α1 · · ·αn))

(5.3) iψα =∑

n

(HD)ψ)α; HD = −iγ0γi∇i + mγ0

The guidance condition is

(5.4)dxµ

n

dt=

ψ†(γ0γµ)nψ

ψ†ψHere the label n enumerates the arguments of the multispinor ψ and the γµ

n act onthe corresponding spinor index αn; ψ is chosen to be antisymmetric with respect tointerchange of any pair of its arguments. For integer spin the formulation in whichthe role of configuration variables is played by coordinates seems to be impossible;instead one has to consider the field spatial configurations as fundamental config-uration variables guided by the corresponding wave functionals. For example thewave functional χ[φ(x), t] for a scalar field φ will obey the standard SE for thecase of curved spacetime with guidance equation as indicated (notational gaps arefilled in below)

(5.5) φ(x, t) =δ

δφ(x)S[φ(x), t]

∣∣∣∣φ(x)=φ(x,t)

(here S[φ(x, t] is times the phase of χ[φ(x), t]). The classical limit in the dynam-ics here is achieved for those configuration variables for which the quantum poten-tial becomes negligible and such variables evolve in accord with classical laws (cf.[471]). One can emphasize that the temporal dynamics of the particle coordinatesand bosonic field configurations completely determine the state, be it microscopicor macroscopic. The role of the wave function in all physical situations is also thesame, namely to provide the guidance laws for configuration variables. In orderthat the probabilities of of different measurements coincide with those calculated

5. PILOT WAVE COSMOLOGY 171

in the standard approach it is necessary to assume that the configuration variablesof the system in a pure quantume ensemble are distributed in accord with the rulep(x) = |ψ(x)|2 (quantum equilibrium condition - cf. [954, 327], Section 2.5, andreferences to Durr, Goldstein, Holland, Valentini, Zanghi, et al for discussion).

Now for quantum gravity recall the ADM formulation for bosonic fields

(5.6) I =∫

M

d3xdt(πabgab + πΦΦ−NH−NaHa)

(here Φ is the set of bosonic fields). The restraints of GR are

(5.7) H =12µ

Gabcdπabπcd + µ

√g(2Λ− 3

R) + HΦ ≈ 0; Ha ≡ −2∇bπ

ba + H

Φa ≈ 0

(only the gravitational parts of the constraints are explicitly written out). Hereµ = (16πG)−1 with G the Newton constant and

(5.8) Gabcd =1√

g(gacgbd + gadgbc − gabgcd)

while ∇a ∼ covariant derivative relative to gab and 3R is the scalar curvature forthe metric g. The classical equations of motion for g are

(5.9) gab =N

µGabcdπ

cd +∇aNb +∇bNa

Recall that in the Schrodinger representation the GR quantum system will be de-scribed by the wave functional Ψ[gab(x,Φ(x), t] over a manifold Σ with coordinatesx and the quantum constraint equations are HµΨ = 0 (H refers to all the com-ponents mentioned above (operator ordering and regularization are not treated in[881]). Putting in now Ψ = Rexp(iS/) one arrives at

(5.10)12µ

δS δS + µ√

g(2Λ− 3R)− 2

2µ

δ δR

R+(Ψ†HΨ)

R2= 0;

δ (R2δS)− 2µ

(Ψ†

HΦΨ) = 0

(here δ ∼ δ/δgab(x) and means contraction with respect to Wheeler’s supermetric(5.8)). Note that for → 0 the first equation in (5.10) reduces to the classicalEinstein-HJ equation. Via the general guidance rules the quantum evolution ofthe gab is now given by (5.9) with the substitution

(5.11) πab(x) → δS

δgab(x)

∣∣∣∣gab(x)=gab(x,t)

The Lagrange multipliers N and Na in (5.9) remain undetermined and are tobe specified arbitrarily. This is analogous to the classical situation where thisarbitrariness reflects reparameterization freedom. Thus to get a solution gab andΦ depending on (x, t) one must first solve the constraint equation HµΨ = 0,then specify the initial configuration (e.g. at t = 0) for gab and Φ, then specifyarbitrarily N(x, t) and Na(x, t), and then solve the guidance equations (5.9) andthe analogous equations for Φ. The solution should then represent a 4-geometryfoliated by spatial hypersurfaces Σ(t) on which the 3-metric is gab(x, t), the lapse


function is N(x, t), the shift vector is Na(x, t) and the field configuration is Φ(x, t).This would be lovely but unfortunately there are complications as indicated belowfrom [770, 772]; such matters are partially anticipated in [881] however and thereis some discussion and calculation. The question of quantum randomness in pilotwave QM is picked up again in the first paper of [881] along with a continuationof time considerations. We only remark here that time in (5.9) is just a universallabel of succession for spatial field configurations; it is not an observable.

5.1. EUCLIDEAN QUANTUM GRAVITY. The discussion here re-volves around [770] and the first paper of [772] (cf. also [263, 264, 266, 218]).We go to [770] directly and refer to [772] for some background calculations andphilosophy. [770] is a review paper of deBroglie-Bohm theory in quantum cos-mology. Extracting liberally one can argue convincingly against the Copenhageninterpretation of quantum phenomena in cosmology, in particular because it im-poses the existence of a classical domain (cf. [729]). Decoherence is discussedbut this does not seem to be a complete answer to the measurement problem(cf. [551, 412, 770]) and one can also argue against the many-worlds theory (cf.[320, 471, 770]). Thence one goes to deBroglie-Bohm as in [84, 881, 954, 961]etc. and the quantum potential enters in a natural manner as we have alreadyseen. Let us follow the notation of [770] here (with some repetition of other dis-cussions) and write H =

∫d3x(NH + N jHj) where (in standard notation) for GR

with a scalar field φ

(5.12) Hj = −2Diπijπφ∂jφ; H = κGijkπ

ijπk +12h−1/2π2

φ+

+h1/2

[−κ−1(R(3) − 2Λ) +

12hij∂iφ∂jφ + U(φ)

]The canonical momentum is expressed via (we use πij instead of Πij)

(5.13) πij = −h1/2(Kij − hijK) = Gijk(hk −DkN −DNk);

Kij = − 12N

(hij −DiNj −DjNi)

K is the extrinsic curvature of the 3-D hypersurface Σ in question with indiceslowered and raised via the surface metric hij and its inverse). The canonicalmomentum of the scalar field is

(5.14) πφ =h1/2

N(φ−N j∂jφ)

As usual R(3) is the intrinsic curvature of the hypersurfaces and N, Nj are thestandard Lagrange multipliers for the super-Hamiltonian constraint H ≈ 0 andthe super momentum constraint Hi ≈ 0. These multipliers are present due to theinvariance of GR under spacetime coordinate transformations. Recall also

(5.15) Gijk =12h1/2(hikhj + hihjk − 2hijhk);

Gijk =12h−1/2(hikhj + hihjk − hijhk)


(called the deWitt metric). Here Di is the i-component of the covariant derivativeon the hypersurface and κ = 16πG/c4. The classical 4-metric

(5.16) ds2 = −(N2 −N iNi)dt2 + 2Nidxidt + hijdxidxj

and the scalar field which are solutions of the Einstein equations can be obtainedfrom the Hamilton equations

(5.17) hij = hij ,H; πij = πij ,H; φ = φ,H; πφ = πφ,Hfor some choice of N, N j provided initial conditions are compatible with the con-straints H ≈ 0 and Hj ≈ 0. It is a feature of the Hamiltonian structure that the4-metrics (5.6) constructed in this way with the same initial conditions describethe same 4-geometry for any choice of N and N j . The algebra of constraints closesin the following form (cf. [470])

(5.18) H(x),H(x′) = Hi(x)∂iδ

3(x, x′)− Hi(x′)∂iδ(x′, x);

Hi(x),H(x′) = H(x)∂iδ(x, x′);

Hi(x),Hj(x′) = Hi(x)∂jδ3(x, x′) + Hj(x′)∂iδ

3(x, x′)

One quantizes following Dirac to get Hi|ψ >= 0 and H|ψ >= 0 and in the metricand field representation the first equation is

(5.19) −2hiDjδψ(hij , φ)

δhj+

δψ(hijφ)δφ

∂iφ = 0

which implies that the wave functional ψ is invariant under space coordinate trans-formations. The second equation is the WDW equation which (in unregularizedform) is

(5.20)−2

[κGijk

δ

δhij

δ

δhk+

12h−1/2 δ2

δφ2

]+ V

ψ(hij , φ) = 0;

V = h1/2

[−κ−1/2(R(3) − 2Λ) +


](V is the classical potential). This equation involves products of local operatorsat the same point and hence must be regulated (cf. also Section 4.4). After thatone should find a factor ordering which makes the theory free of anomalies, in thesense that the commutator of the operator version of the constraints closes in thesame way as their respective classical Poisson brackets (cf. [476, 572, 616]).

Consider now the dBB interpretation of (5.19)-(5.20) where ψ = Aexp(iS/)with A, S functionals of hij and φ. One arrives at(5.21)

−2hiDjδS(hijφ)

δhj+

δS(hij , φ)δφ

∂iφ = 0; −2hiDjδA(hij , φ)

δhj+

δA(hij , φ)δφ

= 0

upon writing ψ = Aexp(iS/). These equations will depend on the factor ordering;however in any case one of the equations will have the form

(5.22) κGijkδS

δhij

δS

δhk+

12h−1/2

(δS

δφ

)2

+ V + Q = 0


where V is the classical potential. Contrary to the other terms in (5.22), whichare already well defined, the precise form of Q depends on the regularization andfactor ordering which are prescribed for the WDW equation. In the unregulatedform of (5.20)

(5.23) Q = −2

A

(κGijk

δ2A

δhijδhk+

h−1/2

2δ2A

δφ2

)The other equation (in addition to (5.22)) is

(5.24) κGijkδ

δhij

(A2 δS

δhk

)+

12h−1/2 δ

δφ

(A2 δS

δφ

)= 0

Now consider the dBB interpretation. First (5.21) and (5.22), which are validirrespective of any factor ordering, are like the HJ equations for GR supplementedby an extra term Q for (5.22) (which does depend on factor ordering etc.). Onepostulates that the 3-metric, the scalar field, and their canonical momenta alwaysexist (independent of observation) and that the evolution of the 3-metric and scalarfield can be obtained from the guidance relations

(5.25) πij =δS(hab, φ)

δhij; πφ =

δS(hij , φ)δφ

(cf. (5.13)-(5.14)). The evolution of these fields will be different from the classicalone due to the presence of Q in (5.22). The only difference between the casesof the nonrelativistic particle and QFT in flat spacetime is the fact that (5.24)for canonical QG cannot be interpreted as a continuity equation for a probabilitydensity A2 because of the hyperbolic nature of the deWitt metric Gijk. Howevereven without a notion of probability density one can extract a lot of informationfrom (5.22), whatever Q may be. First note that whatever the form of Q it mustbe a scalar density of weight one (via the HJ equation (5.22)). From this equationone can express Q via

(5.26) Q = −κGijkδS

δhij

δS

δhk− 1

2h−1/2

(δS

δφ

)2

− V

Since S is an invariant (via (5.21)) it follows that δS/δhij and δS/δφ must be asecond rank tensor density and a scalar density respectively, both of weight one.When their products are contracted with Gijk and multiplied by h−1/2 they forma scalar density of weight one. As V is also a scalar density of weight one thenso must be Q. Further Q must depend only on hij and φ because it comes fromthe wave functional which depends only on these variabes. Of course it can benonlocal but it cannot depend on the momenta.

A minisuperspace is the set of spacelike geometries where all but a set of hij(n)(t)

and the corresponding momenta π(n)ij (t) are put identically equal to zero (this vio-

lates the uncertainty principle but one hopes to retain suitable qualitative features- cf. [770] for references). In the case of a minisuperspace of homogeneous models,one puts Hi = 0 and Ni in H can be set equal to zero without losing any of theEinstein equations. The Hamiltonian becomes HGR = N(t)H(pα(t), qα(t)) where


the qα and pα represent homogeneous degrees of freedom coming from πij(x, t)and hij(x, t). Equations (5.23)-(5.25) become then

(5.27)12fαβ(qµ)

∂S

∂qα

∂S

∂qβ+ U(qµ) + Q(qµ) = 0;

Q(qµ) = − 1R

fαβ∂2R

∂qα∂qβ; pα =

∂S

∂qα= fαβ 1

N

∂qβ

∂t= 0

where fαβ(qµ) and U(qµ) are the minisuperspace particularizations of Gijk and−h1/2R(3)(hij). The last equation is invariant under time reparametrization andhence even at the quantum level different choices of N(t) yield the same spacetimegeometry for a given nonclassical solution qα(x, t).

After some discussion and computations involving the avoidance of singulari-ties and quantum isotropization of the universe one goes now in [770] to the generalsituation in full superspace. From the guidance equations (5.25) one obtains

(5.28) hij = 2NGijkδS

δhk+ DiNj + DjNi; φ = Nh−1/2 δS

δφ+ N i∂iφ

The question here is, given some initial 3-metric and scalar field, what kind ofstructure do we obtain upon integration in t? In particuar does this structureform a 4-dimensional geometry with a scalar field for any choice of the lapse andshift functions? Classically all is well but with S a solution of the modified HJequation containing Q there is no guarantee. One goes to the Hamiltonian picturebecause strong results have been obtained from that point of view historically (cf.[470, 926]). One constructs now a Hamiltonian formalism consistent with theguidance relations (5.25) which yields the Bohmian trajectories (5.28). Given thisHamiltonian one can then use results from the literature to obtain new results forthe dBB picture of quantum geometrodynamics. Thus from (5.21)-(5.22) one caneasily guess that the Hamiltonian which generates the Bohmian trajectories, oncethe guidance relations (5.25) are satisfied initially, should be

(5.29) HQ =∫

d3x[N(H + Q) + N i

Hi

]; HQ = H + Q

Here H and Hi are the usual GR quantities from (5.12) and in fact the guidancerelations (5.25) are consistent with the constraints HQ ≈ 0 and Hi ≈ 0 becauseS satisfies (5.21)-(5.22). Furthermore they are conserved by the Hamiltonian evo-lution given by (5.29). Then one can show that indeed (5.28) can be obtainedfrom HQ with the guidance relations (5.25) viewed as additional constraints (cf.[769, 772] for details). Thus one has a Hamiltonian HQ which generates theBohmian trajectories once the guidance relations (5.25) are imposed initially. Nowone asks about the evolution of the fields driven by HQ forms a 4-geometry as inclassical geometrodynamics. First recall a result from [926] which shows that ifthe 3-geometries and field configurations defined on hypersurfaces are evolved bysome Hamiltonian with the form H =

∫d3x(N H + N iHi) and if this motion can

be viewed as the motion of a 3-dimensional cut in a 4-dimensional spacetime then


the constraints H ≈ 0 and Hi ≈ 0 must satisfy the algebra

(5.30) H(x), H(x′) = −ε[Hi(x)∂iδ3(x′, x)− H(x′)∂iδ

3(x, x′);

Hi(x), H(x′) = H(x)∂iδ3(x, x′);

Hi(x), Hj(x′) = Hi(x)∂jδ3(x, x′)− Hj(x′)∂iδ

3(x, x′)

The constant ε can be ±1 (if the 4-geometry is Euclidean (+1) or hyperbolic (−1));these are the conditions for the existence of spacetime. For ε = −1 this algebra isthe same as (5.18) for GR, but the Hamiltonian (5.29) is different via Q in HQ.The Poisson bracket Hi(x),Hj(x′) satisfies (5.30) because the Hi of HQ definedin (5.29) is the same as in GR. Also Hi(x),HQ(x′) satisfies (5.30) because Hi

is the generator of spatial coordinate transformations and since HQ is a scalardensity of weight one (recall Q is a scalar density of weight one) it must satisfythis Poisson bracket relation with Hi. What remains to be verified is whether thePoisson bracket HQ(x),HQ(x′) closes as in (5.30). For this one recalls a resultof [470] where there is a general super Hamiltonian H which satisfies (5.30), is ascalar density of weight one (whose geometrical degrees of freedom are given onlyby hij and its canonical momentum), and which contains only even powers andno non-local term in the momenta. From [470] this means that H must have theform

(5.31) H = κGijkπijπk +

12h−1/2π2

φ + VG;

VG = −εh1/2

[−κ−1(R(3) − 2Λ) +


]Note that HQ satisfies the hypotheses since it is quadraic in the momenta and thequantum potential does not contain any nonlocal term in the momenta. Conse-quently there are three possible scenarios for the dBB quantum geometrodynam-ics, depending on the form of the quantum potential. First assume that quantumgeometrodynamics is consistent (independent of the choice of lapse and shift func-tions) and forms a nondegenerate 4-geometry. Then HQ,HQ must satisfy thefirst equation in (5.30) and consequently, combining with (5.30) for H, Q must besuch that V + Q = VG with V given by (5.20) yielding(5.32)

Q = −h1/2

[(ε + 1)

(−κ−1R(3) +

12hij∂iφ∂jφ

)+

2κ

(εΛ + Λ) + εU(φ) + U(φ)]

For this situation there are two possibilities, namely(1) The spacetime is hyperbolic with ε = −1 and

(5.33) Q = −h1/2

[2κ

(Λ− Λ)− U(φ) + U(φ)]

Hence Q is like a classical potential; its effect is to renormalize the cos-mological constant and the classical scalar potential, nothing more. Thequantum geometrodynamics is indistinguishable from the classical one.It is not necessary to require Q = 0 since VG = V + Q may alreadydescribe the classical universe in which we live.

6. BOHM AND NONCOMMUTATIVE GEOMETRY 177

(2) The spacetime is Euclidean with ε = 1 in which case

(5.34) Q = −h1/2

[2(−κ−1R(3) +

12hij∂iφ∂jφ

)+

2κ

(Λ + Λ) + U(φ) + U(φ)]

Now Q not only renormalizes the cosmological constant and the classicalscalar field but also changes the signature of spacetime. The total po-tential VG = V + Q may describe some era of the early universe when ithad Euclidean signature but not the present era when it is hyperbolic.The transition between these two phases must happen in a hypersur-face where Q = 0 which is the classical limit. This result points in thedirection of [410].

There remains the possiblity that the evolution is consistent but does not forma nondegenerate 4-geometry. In this case the Poisson bracket HQ,HQ does notsatisfy (5.30) but is weakly zero in some other manner. Consider for example

(1) For real solutions of the WDW equation, which is a real equation, thephase S is zero and from (5.22) one can see that Q = −V . Hence thequantum super Hamiltonian (5.29) will contain only the kinetic term andHQ,HQ = 0 (strong equality). This case is connected with the stronggravity limit of GR (cf. [770] for references and further discussion).

(2) Any nonlocal quantum potential breaks spacetime and as an exampleconsider Q = γV where γ is a function of the functional S (hence isnonlocal). Calculating one obtains (cf. [769, 772])

(5.35) HQ(x),HQ(x′) = (1 + γ)[Hi(x)∂iδ3(x, x′)− H

i(x′)∂iδ3(x′, x)]−

− dγ

dSV (x′)[2HQ(x)− 2κGijk(x)πij(x)Φk(x)− h−1/2πΦ(x)Φφ(x)+

+dγ

dSV (x)[2HQ(x′)− 2κGijk(x′)πij(x′)Φk(x′)− h−1/2πφ(x′)Φφ(x′)] ≈ 0

The last equation is weakly zero because it is a combination of the con-straints and the guidance relations of Bohmian theory. This means thatthe Hamiltonian evolution with the quantum potential Q = γV is con-sistent only when restricted to the Bohmian trajectories. For other tra-jectories it is inconsistent.

In these examples one makes contact with the structure constants of the algebracharacterizing the foam like pregeometry structure pointed out long ago by J.A.Wheeler. Another fact here of interest is that there are no inconsistent Bohmiantrajectories (cf. [772]). We call attention also to [77] where in particular oneconsiders noncommutative geometry and cosmology in connection with Bohmiantheory; the results are very interesting.

6. BOHM AND NONCOMMUTATIVE GEOMETRY

We extract here from [77, 78, 399] with other references as we go along (cf.in particular [230, 261, 318, 341, 552, 771, 772, 773, 897, 901]). First from[77] one refers to [78] where a new interpretation of the canonical commutation

(6.1) [Xµ, Xν ] = iθµν

was proposed. The idea was that it is possible to interpret the commutationrelation as a property of the particle coordinate observables rather than of thespacetime coordinates and this enforced a reinterpretation of the meaning of thewave function in noncommutative QM (NCQM). In [77] one develops a Bohmianinterpretation for NCQM amd forms a deterministic theory of hidden variablesthat exhibit canonical noncommutativity (6.1) between the particle position ob-servables. There are several motivations for reconsideration of hidden variabletheory (see e.g. [475]) and we begin with a Moyal star product defined via

(6.2) (f ∗ g) =1

(2π)n

∫dmkdnpei(kµ+pµ)xµ−(1/2)kµθµνpν f(k)g(p) =

= e(1/2)θµν(∂/∂ξµ)(∂/∂ην)f(x + ξ)g(x + η)|ξ=η=0

(cf. [192] for an extensive treatment of star products and some noncommutativegeometry). The commutative coordinates xi are called the Weyl symbols (WS)of position operators Xi and one will consider them as spacetime coordinatesfollowing [78]. One assumes here that θ0i = 0 and the Hilbert space of states ofNCQM can be taken as in commutative QM with noncommutative SE now givenby

(6.3) i∂ψ(xi, t)

∂t= − 2

2m∇2ψ(xi, t) + V (xi) ∗ ψ(xi, t) =

=2

2m∇2ψ(xi, t) + V

(xj + i

θjk

2∂k

)ψ(xi, t)

The operators

(6.4) Xj = xj +iθjk∂k

2are the observables corresponding to the physical positions of the particles andxi are the associated canonical coordinates. For intition one could think here interms of a “half dipole” whose extent is proportional to the canonical momentum∆xi = θijpj/2; one of its endpoints carries the mass and is responsible for itsinteraction. The NCQM formulated with (6.3)-(6.4) can be considered as theuaual QM with a Hamiltonian not quadratic in momenta and “unusual” positionoperators. From this point of view the BNCQM below can be considered as anextension along the same lines. Any attempt to localize the particles must satisfythe uncertainty relations

(6.5) ∆Xi∆Xj ≥ |θij |/2

The expression for the definition of probability density ρ(xi, t) = |ψ(xi, t)|2 hasa meaning that differs from ordinary QM. Namely the quantity ρ(xi, t)d3x mustbe interpreted as the probability that the system is found in a configuration suchthat the canonical coordinate of the particle is contained in a volume d3x aroundthe point x at time t. Given an arbitrary physical observable characterized by aHermitian operator A(xi, pi) (which includes e.g. A(Xi(xi, pi), pi)) its expectedvalue is

(6.6) < A >t=∫

d3xψ∗(xi, t)A(xj ,−i∂j)ψ(xi, t)

A HJ formalism for NCQM is found by writing ψ = Rexp(iS/), putting it in(6.3), and splitting real and imaginary terms; the real part is

(6.7)∂S

∂t+

(∇S)2

2m+ V + Vnc + QK + QI = 0

Here the three new potential terms are defined as

(6.8) Vnc = V

(xi − θij

2∂jS

)− V (xi);

QK = (− 2

2m

∇2ψ

ψ

)−(

2

2m(∇S)2

)= − 2

2m

∇2R

R;

QI = (

V [xj + (iθjk/2)∂k]ψψ

)− V

(xi − θij

2∂jS

)Vnc is the potential that accounts for the NC classical interactions while QK andQI account for the quantum effects. The NC contributions contained in the lattertwo can be split out by defining

(6.9) Qnc = QK + QI −Qc; Qc = − 2

2m

∇2Rc

Rc; Rc =

√ψ∗

cψc

Here ψc is the wave function obtained from the commutative SE containing theusual potential V (xi), i.e. the equation obtained by setting θij = 0 in (6.3) beforesolving. The imaginary part of the SE which yields the probability conservationlaw, is

(6.10)∂R2

∂t+∇ ·

(R2∇S

m

)+ Σθ = 0; Σθ = −2R

[e−iS/V ∗ (ReiS/)

]By integrating the first equation we obtain

(6.11)d

dt

∫R2d3x = 0;

∫Σθd

3x = 0

when R2 vanishes at infinity.

Now for an ontological theory of motion one follows the traditional methods(cf. [126, 471]). Necessary conditions for the theory to be capable of reproducingthe same statistical results as the standard interpretation of NCQM constrain theadmissible form for the functions Xi(t) which eliminates a certain arbitrariness inthe constructions. First for the rules, with an arbitrary physical characterized bya Hermitian operator A(xi, pi) it is possible to associate a function A(xi, t), thelocal expectation value of A (cf. [471]), which when averated over the ensemble ofdensity ρ(xi, t) = |ψ|2 gives the same expectation value obtained by the standardoperator formalism. Thus it is natural to define the ensemble average via

(6.12) < A >t=∫

ρ(xi, t)A(xi, t)d3x

For this to agree with (6.6) A(xi, t) must be defined as

(6.13) A(xi, t) =[ψ∗(xi, t)A(xj ,−i∂j)ψ(xi, t)

]ψ∗(xi, t)ψ(xi, t)

= A(xi, t) + QA(xi, t)


where the real part is taken to account for the hemiticity of A(xi, pi) and A(xi, t) =A[xi, pi = ∂iS(xi, t)] while QA is defined via

(6.14) QA = [

A(xj ,−i∂j)ψ(xk, t)ψ(xi, t)

]−A(xi, t)

and this is the quantum potential that accompanies A(xi, t). From (6.13) one findsthat the local expectation value of (6.4) is

(6.15) Xi = xi − θij

2∂jS(xi, t)

Now to find the Xi(t) the relevant information for particle motion can be extractedfrom the guiding wave ψ(xi, t) by first computing the associated canonical positiontracks xi(t) and then evaluating (6.15) at xi = xi(t). In order to find a goodequation for the xi(t) it is interesting to consider the Heisenberg formulation andthe equations of motion for the observables. Thus for the xi one has

(6.16)dxi

H

dt=

1i

[xiH , H] =

piH

m+

θij

2

∂V (XkH)

∂XjH

By passing the right side of (6.16) to the Schrodinger picture one can define thevelocity operators

(6.17) vi =1[xi, H] =

pi

m+

θij

2

∂V (X)∂X

The differential equation for the canonical positions xi(t) is now found by identi-fying dxi(t)/dt with the local expectation value of vi, thus

(6.18)dxi(t)

dt=[∂iS(xi, t)

m+

θij

2

∂V (Xk)∂Xj

+Qi

2

]∣∣∣∣xi=xi(t)

where Xi is given in (6.15), S(xi, t0 is the phase of ψ, and

(6.19) Qi =

((θij/)[∂V (Xk)/∂Xj ]ψ(xi, t)

ψ(xi, t)

)− θij

∂V (Xk)∂Xj

The potentials Qi account for quantum effects coming from derivatives of order 2and higher contained in ∂V (Xi)/∂Xj . Then once the xi(t) are known the particletrajectories are given via

(6.20) Xi(t) = xi(t)− θij

2∂jS(xk(t), t)

One notes that the particles positions are not defined on nodal regions of ψ, whereS is undefined, so the particles cannot run through such regions. Hence the van-ishing of the wave function can be adopted as a boundary condition, implying thatthe particle does not run through such a region (see [77] for more discussion). Thepreceeding theory is now summarized in a formal structure as follows:

(1) The spacetime is commutative and has a pointwise manifold structurewith canonical coordinates xi. The observables correspondin to opera-tors of position coordinates Xi of particles satisfy the commutation rules


[Xk, Xj ] = iθkj . The position observables can be represented in the coor-dinate space as Xj = xj + iθjk∂k/2 and the xj are canonical coordinatesassociated with the particle.

(2) A quantum system is composed of a point particle and a wave ψ. Theparticle moves in spacetime under the guidance of the wave which satisfiesthe SE i∂tψ(xi, t) = −(2/2m)∇2ψ + V (Xi)ψ (ψ = ψ(xi, t)).

(3) The particle moves with trajectory Xi(t) = xi(t)− (θij/2)∂jS(xi(t), t)independently of observation, where S is the phase of ψ and the xi(t)describe the canonical position trajectories found by solving

dxi(t)dt

=[∂iS(x, t)

m+

θij

2

∂V (Xk)∂Xj

+Qi

2

]∣∣∣∣xi=xi(t)

To find the path followed by a particle, one must specify its initial canoni-cal position xi(0), solve the second equation, and then obtain the physicalpath from the first equation.

These three postulates constitute on their own a consisent theory of motion, andis intended to give a finer view of QM, namely a detailed description of the in-dividual physical processes and to provide the same statistical predictions. Inordinary commutative Bohmian mechanics, in order to reproduce the statisticsthe additional requirement that ρ(xi, t0) = |ψ(x, t0)|2 is imposed for some initialtime t0. Then ρ is said to be equivariant if this property persists under evolutionxi(t) = f i(xj , t); in such a case

(6.21)∂ρ

∂t+

∂(ρxi)∂xi

= 0

In ordinary commutative QM the equivariance property is satisfied via xi(t) =J i/ρ which is a consequenc of the identification between xi and the local expecta-tion value of the vi. In the BNCQM proposed here the same identification is validbut it is not sufficient to guarantee equivariance in all cases. This is clear aftercomputing the canonical probability current

(6.22) J i(xi, t) = [ψ∗vψ] = |ψ|2[∂iS(x, t)

m+

θij

2

∂V (Xk)∂Xj

+Qi

2

]= ρxi

Then regrouping the terms in (6.10) so that the canonical probability flux (6.22)appears explicitly one obtains

(6.23)∂ρ

∂t+

∂(ρxi

∂xi− ∂

∂xi

[ρ

(θij

2

∂V (Xk)∂Xj

+Qi

2

)]+ Σθ = 0

For equivariance to occur an additional condition that the sum of the last twoterms in the right side of (6.23) vanishes is required. When V (Xi) is a linear orquadratic function, as in many applications, such a condition is trivially satisfied,and then ρ(xi, t) = |ψ(xi, t)|2 as desired. The same may occur for other situationswhen other potentials are considered in #2 above and this is also discussed in [77].

In [77] (second paper) one looks at a Kantowski-Sachs (KS) universe (see e.g.


[230, 261, 399, 901]). Recall in the ADM formulation a line element is writtenin the form

(6.24) ds2 = (NiNi −N2)dt2 + 2Nidxidt + hijdxidxj

and the Hamiltonian of GR without matter is

(6.25) H =∫

d3x(NH + N)iHj); H = Gijkπijπk − h1/2R(3); H

j = 2Diπij

Units are chosen so that = c = 16πG = 1. The momenta πij are canonicallyconjugate to hij and the deWitt metric Gijk are given via

(6.26) πij = −h1/2(Kij − hijK); Gijk =12h−1/2(hikhj + hihjk − hijhk)

where Kij = −(∂thij −DiNj −DjNi)/(2N) is the second fundamental form. Thesuper Hamiltonian constraint H ≈ 0 yields the WDW equation

(6.27)(

Gijk δ

δhij

δ

δhk+ h1/2R(3)

)ψ[hij ] = 0

In the Bohmian approach now one has

(6.28) πij = −h1/2(Kij − hijD) =

1ψ∗ψ

[ψ∗

(−i

δ

δhij

)ψ

]=

δS

δhij

If one puts ψ = Aexp(iS) in (6.27) there results

(6.29) Gijk δS

δhij

δS

δhk− h1/2R(3) + Q = 0;

Gijk δS

δhij

(A2 δS

δhk

)= 0; Q = − 1

AGijk δ2A

δhijhk

(one should really include 2 here in dealing with Q).

The Kantowski-Sachs universe is an important anisotropic model; the lineelement is

(6.30) ds2 = −Ndt2 + X2(t)dr2 + Y 2(t)(dθ2 + Sin2(θ)dφ2)

In the Misner parametrization this becomes (cf. [399])

(6.31) ds2 = −N2dt2 + e2√

3βdr2 + e−2√

3βe−2√

3Ω(dθ2 + Sin2(θ)dφ2)

The Hamiltonian is then

(6.32) H = NH = Nexp(√

3β + 2√

3Ω)[−P 2

Ω

24+

P 2β

24− 2exp(−2

√3Ω)

]One sets Θ = V α

;α (volume expansion - V α = δα0 /N), and σ2 = σαβσαβ/2 (shear)

where σαβ = (hµαhν

β+hµβhν

α)Vµ;ν/2. The semicolon denotes 4-dimensional covariantdifferentiation and hµ

α = δµα +V µVα is the projector orthogonal to the observer V α

(cf. [458]). A characteristic length scale is defined via Θ = 3/(N) and in thegauge N = 24exp(−

√3β − 2

√3Ω) one has

(6.33) Θ(t) =1N

(X

X+ 2

Y

Y

)= −

√3

24(β + 2Ω)e

√3β+2

√3Ω;


σ(t) =1

N√

3

(X

X− Y

Y

)=

124

(2β + Ω)e√

3β+2√

3Ω;

3(t) = X(t)Y 2(t) = e−√

3β(t)−2√

3Ω(t)

In order to distinguish the role of the quantum and noncommutative effects ina NC quantum universe one starts now with a KS geometry in the commutativeclassical version. The Poisson brackets for the classical phase space variables are

(6.34) Ω, PΩ = 1 = β, Pβ; PΩ, Pβ = 0 = Ω, β

For the metric (6.31) the constraint H ≈ 0 is reduced to

(6.35) H = ξh ≈ 0; ξ =124

e√

3β+2√

3Ω; h = −P 2Ω + P 2

β − 48e−2√

3Ω ≈ 0

The classical equations of motion for the phase space variables Ω, PΩ, β, and Pβ

are then

(6.36) Ω = NΩ,H = −2PΩ; β = Nβ,H = 2Pβ ;

PΩ = NPΩ,H = −96√

3e−2√

3Ω; Pβ = NPβ ,H = 0

Explicit formulas are found and exhibited in [77].

Now for a NC classical model one considers

(6.37) Ω, PΩ = 1; β, Pβ = 1; PΩ, Pβ = 0; Ω, β = θ

The equations of motion can be written as

(6.38) Ω = −2PΩ; PΩ = −96√

3e−2√

3Ω; β = 2Pβ − 96√

3θe−2√

3Ω; Pβ = 0

The solutions for Ω and β are then

(6.39) Ω(t) =√

36

log

48P 2

β0

Cosh2[2√

3Pβ0(t− t0)]

;

β(t) = 2Pβ0(t− t0) + β0 − θPβ0Tanh[2√

3Pβ0(t− t0]

Further calculations appear in [77].

For the commutative quantum model one works with the minisuperspace con-struction of homogeneous models and freezing out an infinite number of degreesof freedom. First an Ansatz of the form (6.31) is introduced and the spatial de-pendence of the metric is integrated out. The WDW equation is then reduced toa KG equation which for the KS universe has the form

(6.40)[−P 2

Ω + P 2β − 48e−2

√3Ω]ψ(Ω, β) = 0

where PΩ = −i∂/∂Ω and Pβ = −i∂/∂β. A solution to (6.40) is then (cf. [399])

(6.41) ψν(Ω, β) = eiν√

3βKiν

(4e−

√3Ω)


where Kiν is a modified Bessel function and ν is a real constant. Once a quantumstate of the universe is given as, e.g. a superposition of states

(6.42) ψ(Ω, β) =∑

ν

Cνeiν√

3βKiν

(4e−

√3Ω)

= ReiS

the evolution can be determined by integrating the guiding equation (6.28). Inthe minisuperspace approach the analogue of that equation is

(6.43) PΩ = −12Ω =

[ψ∗(−i∂Ω)ψ]

ψ∗ψ

=

∂S

∂Ω;

Pβ =12β =

[ψ∗(−i∂β)ψ]

ψ∗ψ

=

∂S

∂β

As before one has fixed the gauge N = 243 = 24exp(−√

3β − 2√

3Ω). Usuallydifferent choices of time yield different quantum theories but when one uses theBohmian interpretation in minisuperspace models the situation is identical to thatof the classical case (but not beyond minisuperspace - cf. [772]), namely differentchoices yield the same theory (cf. [84]). Hence as long as 3(t) does not passthrough zero (a singularity) the above choice for N(t) is valid for the history ofthe universe. The minisuperspace analogue of the HJ equation in (6.29) is

(6.44) − 124

(∂S

∂Ω

)2

+124

(∂S

∂β

)2

− 2e−2√

3Ω +1

24R

(∂2R

∂Ω2− ∂2R

∂β2

)= 0

Explicit calculations are then given in [77] with graphs and pictures.

Now for the NC quantum model one takes

(6.45) [Ω, β] = iθ

According to the Weyl quantization procedure (cf. [318]) the realization of (6.45)in terms of commutative functions is made by the Moyal star product defined via(cf. [192])

(6.46) f(Ωc, βc) ∗ g(Ωc, βc) = f(Ωc, βc)ei(θ/2)(←−∂ Ωc

−→∂ βc−

←−∂ βc

−→∂ gOc )g((Ωc, βc)

The commutative coordinates Ωc, βc are called Weyl symbols of the operatorsΩ, β. In order to compare evolutions with the same time parameter as above oneagain fixes the gauge N = 24exp(−

√3β − 2

√3Ω) and the WDW equation for the

NC KS model is (cf. [399])

(6.47)[−P 2

Ωc+ P 2

βc− 48e−2

√3Ωc

]∗ ψ(Ωc, βc) = 0

which is the Moyal deformed version of (6.40). By using properties of the Moyalbracket (cf. [192]) one can write the potential term (denoted by V to include thegeneral case) as

(6.48) V (Ωc, βc) ∗ ψ(Ωc, βc) = V

(Ωc + i

θ

2∂βc

, βc − iθ

2∂Ωc

)ψ(Ωc, βc) =

= V (Ω, β)ψ(Ωc, βc); Ω = Ωc −θ

2Pβc

; β = βc +θ

2PΩc


The WDW equation then reads as

(6.49)[−P 2

Ωc+ P 2

βc− 48e−2

√3Ωc+

√3θPβc

]ψ(Ωc, βc)

Two consistent interpretations for the cosmology emerging from these equationsare possible. One consists in considering the Weyl symbols Ωc and βc as theconstituents of the physical metric, which makes things essentially commutativewith a modified interaction. The second, as adopted in [78, 79], involves theWeyl symbols being considered as auxiliary coordinates, and thereby one studiesthe evolution of a NC quantum universe.

Next for the Bohmian formulation of the NC minisuperspace one looks at

(6.50) [xi, pj ] = iδij

Note the operator formalism of QM is not a primary concept in Bohmian mechan-ics. Thus it is reasonable to expect that in Bohmian NC quantum cosmology itshould be possible to describe the metric variables as well defined entities, althoughthe operators Ω and β satisfy (6.45). This is exactly what is proposed here. Onewants to give an objective meaning to the wavefunction and the metric variablesΩ and β and the wave function is obtained by solving (6.47). What is missing isthe evolution law for Ω and β. To find this one associates a function A(Ωc, βc) toA(Ωc, βc, PΩc

, Pβc) according to the rule

(6.51) B[A] =[ψ∗(Ωc, βc)A(Ωc, βc,−i∂Ωc

,−i∂βc)ψ(Ωc, βc)]

ψ∗(Ωc, βc)ψ(Ωc, βc)= A(Ωc, βc)

where the real part takes into account the hermiticity of A (the B here refers tothe idea of “beable”). Applying this to the operators Ω and β one arrives at

(6.52) Ω(Ωc, βc) = B[Ω] =[ψ∗(Ωc, βc)Ω(Ωc,−i∂βc

)ψ(Ωc, βc)]ψ∗(Ωc, βc)ψ(Ωc, βc)

= Ωc −θ

2∂βc

S;

β(Ωc, βc) = B[β] =[ψ∗(Ωc, βc)β(βc,−i∂Ωc

)ψ(Ωc, βc)]ψ∗ψ

= βc +θ

2∂Ωc

S

Thus the relevant information for universe evolution can be extracted from theguiding wave ψ(Ωc, βc) by first computing the associated canonical position tracksΩc(t) and βc(t). Then one obtains Ω(t) and β(t) by evaluating (6.52) at Ωc = Ωc(t)and βc = βc(t) (similar procedures are worked out in [77] for the NC classical sit-uation). Differential equations for the canonical positions Ωc(t) and βc(t) can befound by identifying Ωc(t) and βc(t) with the beables associated with their timeevolution and formulas are worked out in [77].

The combination of NC geometry and Bohmian type quantum physics is some-what like a fusion of two apparently opposite ways of thinking, one fuzzy and theother refering ontologically to point particles. Every Hermitian operator can beassociated with an ontological element and by averaging the beable B[A] over anensemble of particles with probability density ρ = |ψ|2 one gets the same resultas computing the expectation value of the observable A in standard operationalformalism. In the KS universe, where WDW is of KG type (with no notion of


probability) the beable mapping is well defined, even in the NC case. In thecommutative context this formulation leads to the Bohmian quantum gravity pro-posed by Holland in [471] in the minisuperspace approximation. The work hereshows that noncommutativity can modify appreciably the universe evolution inthe quantum context (qualitatively as well as quantitively).

We go next to [78] and consider NC in the evolution of Friedman-Robertson-Walker (FRW) universes with a conformally coupled scalar field. First take thecommutative situation and restrict attention to the case of constant positive cur-vature of the spatial sections. The action is then

(6.53) S =∫

d4x√−g

[−1

2φ;µφ;µ +

R

16πG− Rφ2

12

]Units are chosen so that = c = 1 and 8πG = 32P where P is the Plancklength. For the FRW model with a homogeneous scalar field the following Ansatzof minisuperspace can be adopted

(6.54) ds2 = −N2(t)dt2 + a2(t)[

dr2

1− r2+ r2(dθ2 + Sin2(θ)dφ2)

]; φ = φ(t)

Rescaling the scalar field via χ = φaP /√

2 one obtains the minisuperspace action,Hamiltonian, and momenta

(6.55) S =∫

dt

(Na− aa2

N+

aχ2

N− Nχ2

a

); Pa = −2aa

N;

H = N

[−P 2

a

4a+

P 2χ

4a− a +

χ2

a

]= NH; Pχ =

2aχ

N

For the classical phase space variables one knows a, χ = 0 = Pa, Pχ anda, Pa = 1 = χ, Pχ and the equations for the metric and matter field variablesfollowing from this and (6.54) are

(6.56) a = a,H = −NPa

2a; Pa = Pa,H = 2N ;

χ = χ,H =NPχ

2a; Pχ = Pχ,H = −2Nχ

aNow one adopts the conformal time gauge N = a and the general solution of (6.56)in this gauge is

(6.57) a(t) = (A + C)Cos(t) + (B + D)Sin(t);

χ(t) = (A− C)Cos(t) + (B −D)Sin(t)

where the constraint H ≈ 0 imposes the relation AC + BD = 0.

Next look at a NC deformation by keeping the Hamiltonian with the samefunctional form as in (6.55) but now with NC variables

(6.58) H = N

[−

P 2anc

4anc+

P 2χnc

4anc− anc +

χ2nc

anc

]


where

(6.59) anc, χnc = θ; anc, Panc = 1 = χnc, Pχnc

; Panc, Pχnc

= 0

Now make the substitution

(6.60) anc = ac −θ

2Pχc

; χnc = χc +θ

2Pac

; Pac= Panc

, Pχc= Pχnc

Then the theory defined by (6.58)-(6.59) can be mapped to a theory where themetric and matter variables satisfy

(6.61) ac, χc = 0 = Pac, Pχc

; ac, Pac = χc, Pχc

= 1

In the case where ac, χc are taken as the preferred variables one has a commutativetheory referred to as a theory realized in the C-frame (cf. [399]). When anc andχnc are used as constituents of the physical metric and matter field one refersto the NC frame (cf. [77, 78, 79]). Some work assumes the difference betweenthe C and NC variables is negligible (cf. [234]) but as shown in [79] even insimple models the difference in behavior between these two types of variables canbe appreciable; here one shows that the assumption of C or NC frame leads todramatic differences in the analysis of universe history. Calculations are made forthe classical situation in both sets of variables which we omit here.

Next comes the quantum version of the commutative universe model for theFRW universe with conformally coupled scalar field; this has been investigatedon the basis of the WDW equation in e.g. [108] using Bohmian trajectories butthere was there a restriction to the regime of small scale parameters and thewavefunctions there were different from those used here. One writes then Pa =−∂/∂a and Pχ = −i∂/∂χ to get from (6.55) the WDW equation

(6.62)[− ∂2

∂a2+

∂2

∂χ2+ 4(a2 − χ2)

]ψ(a, χ) = 0

One can separate variables as in [108, 552] but here one chooses a different routemore suitable for application in the NC situation. Thus write a = ξCosh(η) andχ = ξSinh(η) to get

(6.63)[(

∂2

∂ξ2+

1ξ

∂

∂ξ− 1

ξ2

∂2

∂η2

)− 4ξ2

]ψ(ξ, η) = 0

Putting in ψ = R(ξ)exp(iαη) one obtains

(6.64)∂2R

∂ξ2+

1ξ

∂R

∂ξ+(

α2

ξ2− 4ξ2

)R = 0

A solution is R(ξ) = AKiα/2(ξ2)+BIiα/2(ξ2) where Kν and Iν are Bessel functionsof the second kind, A, B are constants, and α is a real number. The solution ofWDW is then

(6.65) ψ(ξ, η) = AKiα/2(ξ2)eiαη + BIiα/2(ξ2)eiαη

Such wavefunctions appear in the study of quantum wormholes (cf. [230]) and inquantum cosmology for the KS universe (cf. [399, 892]). One often discards the

Iν solution (not always wisely) and uses

(6.66) ψ(ξ, η) =∑α

AαKiα/2(ξ2)eiαη

as a solution.

Now for the Bohmian approach one recalls from quantum information theorythat the wavefunction has a nonphysical character (cf. [758]) and Bohmian theoryshould be in accord with this in some way. In the present picture the object ofattention is the primordial quantum universe, characterized in the minisuperspaceformalism by the configuration variables a and χ. Having fixed the ontologicalobjects one must determine how they evolve in time and this is done with the aidof the wave function. Thus an evolution law is ascribed to point particles via

(6.67) xi =

1m

[ψ∗(−i∂i)ψ]ψ∗ψ

=∇S

m; i∂ψ = − 2

2m∇2ψ + V ψ

where ψ is the wavefunction of the universe and S comes from ψ = Aexp(iS).All phenomena governed by nonrelativistic QM follow from the analysis of thisdynamical system. The expectation value of a physical quantity associated with aHermitian operator A(xi, pi) can be computed in the Bohmian formalism via

(6.68) B(A) =

[ψ∗A(xi,−i∂i)ψ]ψ∗ψ

= A(xi, t)

which represents the same quantity when seen from the Bohmian perspective (cf.[471]). In the context of nonrelativistic QM it can be shown from first principlesthat for an ensemble of particles obeying the evolution law (6.67) (first equation)the associated probability density is ρ = |ψ|2 (cf. [327]). This is why computingthe ensemble average of A(xi, t) via

(6.69)∫

d3xρA(xi, t) =∫

d3xψ∗A(xi,−i∂i)ψ =< A >t

leads to the same result as the standard operator formalism. Note that the lawof motion in (6.67) can itself be obtained from (6.68) by associating xi with thebeable corresponding to the velocity operator, namely

(6.70) xi = B(i[H, xi]) = ∇S/m

Again Bohmian mechanics does not give to probability a privileged role, but, asdiscussed in [327], probability is a derived concept arising from the laws of motionof point particles. In this sense the Bohmian approach is suited to an isolatedsystem (such as the universe) but on the other hand there might be an ensembleof universes.

Now for quantum cosmology on the present context of a FRW universe witha conformally coupled scalar field. In the commutative case the resulting Bohmianminisuperspace formalism matches with the minisuperspace version of the Bohmianquantum gravity proposed in [471] and employed in [108]. From (6.70) one has


in the gauge N = a

(6.71) a =

[ψ∗(−i∂a/2)ψ]ψ∗ψ

= −1

2∂S

∂a; χ =

[ψ∗(−i∂χ/2)ψ]

ψ∗ψ

=

12

∂S

∂χ

Changing into the (ξ, η) coordinates we obtain

(6.72)dξ

dt= −1

2∂S(ξ, η)

∂ξ;

dη

dt=

12ξ2

∂S(ξ, η)∂η

For a single Bessel function in (6.66) one has ψ(ξ, η) = AKiα/2(ξ2)exp(iαη) whereA is a constant. From S = αη the equations of motion in (6.72) reduce to

(6.73)dξ

dt= 0;

dη

dt=

α

2ξ2⇒ ξ = ξ0; η =

α

2ξ20

t + η0

leading to

(6.74) a(t) = ξ0Cosh

(α

2ξ20

+ η0

); χ(t) = ξ0Sinh

(α

2ξ20

+ η0

)Quantum effects can therefore remove the cosmological singularity giving rise tobouncing universes. The case of a superposition of two Bessel functions of typeKiν is also written out in part but the calculations become difficult.

For the NC quantum model one takes the quantum version of (6.59) as

(6.75) [a, χ] = iθ; [a, Pa] = i; [χ, Pχ] = i; [Pa, Pχ] = 0

These commutation relations can be realized in terms of commutative functionsby making use of star products as in (6.46). The commutative coordinates ac, χc

are called Weyl symbols as before and a WDW equation is

(6.76)[P 2

ac− P 2

χc

]ψ(ac, χc) + 4(a2

c − χ2c) ∗ ψ(ac, χc) = 0

(obtained by Moyal deforming (6.62)). The resulting equations are simply operatorversions of (6.60).

For the NC Bohmian version one departs from the C-frame and uses thebeable mapping (6.68) to ascribe an evolution law to the canonical variables. Intime gauge N = anc the Hamiltonian (6.58) reduces to

(6.77) h =

[−

P 2anc

4+

P 2χnc

4− a2

nc + χ2nc

]One can therefore use h to generate time dependence and obtain the Bohmianequations as

(6.78)dac

dt= B(i[h, ac]) = −1

2(1− θ2)

∂S

∂ac+ θχc;

dχc

dt= B(i[g, χc]) =

12(1− θ2)

∂S

∂χc+ θac

The connection between the C and NC frame variables is established b applyingthe “beable” mapping to the operator equations based on (6.60), namely a =ac − (θ/2)Pχc

, etc.; that is by defining a ≡ B(a) and χ = B(χ). Once the


trajectories are determined in the C frame one can find their counterparts in theNC frame by evaluating the variables a and χ along the C-frame trajectories

(6.79) a(t) = B(a)| ac=ac(t)χc=χc(t)

= ac(t)−θ

2∂χc

S[ac(t), χc(t)];

χ(t) = B(χ)| ac=ac(t)χc=χc(t)

= χc(t) +θ

2∂ac

S[ac(t), χc(t)]

Now for an application to NC quantum cosmology use Pac= −i∂ac

and Pχc=

−i∂χcwith NC WDW from (6.76) written as

(6.80)[β

(− ∂2

∂a2c

+∂2

∂χ2c

)+ 4(a2

c − χ2c) + 4iθ

(χc

∂

∂ac+ ac

∂

∂χc

)]ψ(ac, χc) = 0

where β = 1− θ2. Separation of variables works, after writing ac = ξCosh(η) andχc = ξSinh(η), allowing (6.80) to be rewritten as

(6.81)[β

(∂2

∂ξ2+

1ξ

∂

∂ξ− 1

ξ2

∂2

∂η2

)− 4iθ

∂

∂η− rξ2

]ψ(ξ, η) = 0

Using (6.78) one can write the Bohmian equations of motion as

(6.82)dξ

dt= −1

2(1− θ2)

∂S(ξ, η)∂ξ

;dη

dt=

12ξ2

(1− θ2)∂S(ξ, η)

∂η+ θ

(6.79) can be written in the new set of coordinates as

(6.83) anc(t) = ac(t) +θ

2Sinh(η)∂ξS[ξ(t), η(t)]− θ

2ξ−1Cosh(η)∂ηS[ξ(t), η(t)];

χnc(t) = χc(t) +θ

2Cosh(η)∂ξS[ξ(t), η(t)]− θ

2ξ−1Sinh(η)∂ηS[ξ(t), η(t)]

The paper continues with extensive computations for a variety of situations andwe hope to have captured the spirit of investigtion.

7. EXACT UNCERTAINTY AND GRAVITY

We go here to [444] (cf. Remarks 1.1.4 and 1.1.5 along with Section 3.1) and[186, 187, 189, 203] for background (for the original sources see e.g.[446, 447,448, 449, 450, 805, 806, 807]). The theme here is that the exact uncertaintyapproach may be promoted to the fundamental element distinguishing quantumand classical mechanics. Nonclassical fluctuations are addded to the usual de-terministic connection between the configuration and momentum properties of aphysical system. Assuming that the uncertainty introduced to the momentum(i.e. the fluctuation strength) is fully determined by the uncertainty in the con-figuration via the configuration probability density one arrives at QM from CM.We remark that the quantum potential arises from variation of the Fisher metricwith respect to the probability P and this is another significant feature relatingthe quantum potential to fluctuations and indirectly to the Bohmian formulationof QM (cf. [186, 187, 189, 203]). For a quick review of the particle situa-tion let H = (p2/2m) + V (x) be the Hamiltonian for a spinless particle with SEi∂tψ = H(x,−i∇, t)ψ = −(2/2m)∇2ψ + V ψ and recall that the probability

7. EXACT UNCERTAINTY AND GRAVITY 191

density P is specified as |ψ|2. Thus in this canonical approach there is a lot of blackmagic while in the exact uncertainty approach one uses statistical concepts fromthe beginning and the wavefunction and SE are derived rather than postulated.Thus assume an ensemble picture from the beginning (due to uncertainty in theposition) and assume that a fundamental position probability density P followsfrom an action principle involving

(7.1) A =∫

dt

[H +

∫dxP

∂S

∂t

];

∂P

∂t=

δH

δS;

∂S

∂t= −δH

δP

(no ψ is assumed here). One shows that conservation of probability requires H isinvariant under S → S + c and if H has no explicit time dependence then its valueis a conserved quantity corresponding to energy. As an example here consider theclassical ensemble Hamiltonian

(7.2) Hc[P, S] =∫

dxP

[|∇S|22m

+ V

]Then as above

(7.3)∂P

∂t+∇ ·

[P∇S

m

]= 0;

∂S

∂t+|∇S|22m

+ V = 0

This formalism based on an action principle for the position probability densitysuccessfully describes the motion of ensembles of classical particles; moreover it isconsiderably more general (see e.g. [450]). In particular the essential differencebetween classical and quantum ensembles becomes a matter of form, being charac-terized by a simple difference in the forms of the emsemble Hamiltonians Hc andHq.

Thus assume the physical momentum is given via (♣) p = ∇S + f wherethe fluctuation field f vanishes almost everywhere on average. This is not dis-similar from e.g. Nelson’s mechanics where a Brownian motion is attached, orthe approach of scale relativity (cf. [186, 187, 188, 189, 203] for an extensivetreatment of such matters). One will see that such fluctuations introduce inde-terminism at the level of individual particles but not at the ensemble level. Writenow an overline to denote averaging over the fluctuations at a given position sof = 0 and p = ∇S by assumption and the classical ensemble energy is replaced by

(7.4) < E >=∫

dxP [(2m)−1|∇S + f |2 + V ] =

=∫

dxP [(2m)−1(|∇S|2 + 2f · ∇S + f · f) + V ] = Hc +∫

dxPf · f2m

Now one asks whether this modified classical ensemble can be subsumed withinthe general formalism above and this is OK proved that f · f is determined bysome function of P, S and their derivatives, i.e.

(7.5) f · f = α(x, P, S,∇P,∇S, · · · )


In this case one can define a modified ensemble Hamiltonian

(7.6) Hq = Hc +∫

dxPα(x, p, S,∇P,∇S, · · · )

2m

The aim of the exact uncertainty approach is to fix the form of α uniquely andthis is done by requiring first three generally desirable principles to be satisfied(causality, invariance, and independence), plus an exact uncertainty principle, andgiven this the resulting equations of motion are equivalent to the SE for a quantumensemble of particles. This is covered at length in [186, 205, 203] and in [447] forexample and slightly different versions are given here, following [449] for bosonicfields, in order to make a connection with qauntum gravity. The requirements are:(i) The modified ensemble Hamiltonian Hq leads to causal equations of motion (soα cannot depend on second and higher derivatives of P and S) (ii) The respectivefluctuation strengths for noninteracting uncorrelated ensembles are independent(thus f1 · f1 and f2 · f2 are independent of P1 and P2 respectively when P (x) =P1(x2)P2(x2)) (iii) The fluctuations transform correctly under linear canonicaltransformations (thus f → LT f for any invertible linear coordinate transformationx → L−1x). The fourth assumption is: (iv) The strength of the momentumfluctuations α = f · f is determined solely by the uncertainty in position - henceα can only depend on x, P and derivatives. It is shown in the references abovethat these four principles lead to the unique form

(7.7) Hq[P, S] = Hc[P, S] + C

∫dx∇P · ∇P

2mP

where C is a positive universal constant (i.e. having the same value for all en-sembles). Moreover if one sets = 2

√C and ψ =

√Pexp(iS/) the SE results as

above.

Now for gravitational situations we have an ADM metric

(7.8) ds2 = −(N2 − hijNiNj)dt2 + 2Nidxidt + hijdxidxj

where N and Nj refer to lapse and shifts with hij the spatial metric. Considernow the possibility that the configuration of the field is an inherently imprecisenotion, hence requiring a probability functional P [hij ] for its description. Assumethat the dynamics of the corresponding statistical ensemble are generated by anaction principle δA = 0 where A =

∫dt[H +

∫DhP (∂S/∂t)] analogous to (7.1).

Here∫

Dh denotes a functional integral over configuration space and H dependson P [hij ] and its conjugate functional S[hij ]. The equations of motion are then

(7.9)∂P

∂t=

∆H

∆S;

∂S

∂t= −∆H

∆P

where ∆/∆F denotes the variational derivative with respect to the functional F (cf.[449] for details and Remark 5.1). A suitable “classical” ensemble Hamiltonianmay be constructed from knowledge of the classical equations of motion for anindividual field via (cf. [449])

(7.10) Hc[P, S] =∫

DhPH0[hij , δS/δhij ]

where

(7.11) H0[hij , πij ] =

∫dx

[N

(12Gijkπ

ijπk + V (hij))− 2Niπ

ij|j

]where V is the negative of twice the product of the 3-curvature scalar with[det(h)]1/2 and |j denotes the covariant 3-derivative (this is the single field Hamil-tonian). For the ensemble Hamiltonian Hc in (7.10) one has now

(7.12)∂P

∂t+∫

dxδ

δhij(Phij) = 0;

∂S

∂t+ H0[hij , δS/δhij ] = 0;

hij = NGijkδS

δhk−Ni|j −Nj|i

These equations of motion correspond to the conservation of probality with prob-ability flow hij and the HJ equation for an individual gravitational field withconfiguration hij As is well known the lack of conjugate momenta for the lapseand shift components N and Ni places constraints on the classical equations ofmotion. In the ensemble formalism these constraints take the form (cf. [449])

(7.13)δP

δN=

δP

δNi=

∂P

∂t= 0;

(δP

δhij

)∣∣∣∣|j

= 0;

δS

δN=

δS

δNi=

∂S

∂t= 0;

(δS

δhij

)∣∣∣∣|j

= 0

and this corresponds to invariance of the dynamics with respect to the choice oflapse and shift functions and initial time - and to the invariance of P and S underarbitrary spatial coordinate transformations. Applying these constraints to theabove classical equations of motion yields for the Gaussian choice N = 1, Ni = 0the reduced classical equations

(7.14)δ

δhij

(PGijk

δS

δhk

)= 0;

12Gijk

δS

δhij

δS

δhk+ V = 0

Now the exact uncertainty approach can be adapted in a straightforward wayto obtain a modified ensemble Hamiltonian that generates the quantum equationsof motion. It is assumed first that the classical deterministic relation between thefield configuration hij and its conjugate momentum density πij is relaxed to

(7.15) πij =δS

δhij+ f ij

analogous to (♣) where here f ij vanishes on average for all configurations. Thisadds a kinetic term to the average ensemble energy analogous to (7.4) with

(7.16) Hq =< E >= Hx +12

∫DhP

∫dxNGijkf ijfk

Note here that the term in (7.11) linear in the derivative of πij can be integratedby parts to give a term directly proportional to πij , which remains unchangedwhen the fluctuations are added and averaging is performed. Next one fixes the


form of Hq using the same principles of causality, independence, invariance andexact uncertainty used before (cf. [449] for details) leading to

(7.17) Hq[P, S] = Hc[P, S] +C

2

∫Dh

∫dxNGijk

1P

δP

δhij

δP

δhk

analogous to (7.7) where C is a positive constant with the same value for all fields.The corresponding modified equations of motion may be calculated via (7.9) andthe constraints in (7.13) applied to obtain reduced equations analogous to (7.14).If one now defines = 2

√C and Ψ[hij ] =

√Pexp(iS/) these reduced equations

can be rewritten in the form (cf. [449])

(7.18)[−2

2δ

δhijGijk

δ

δhk+ V

]Ψ = 0

This is of course the WDW equation for quantum gravity. Note also that theconstraints in (7.13) may be rewritten in terms of the wavefunctional Ψ as

(7.19)δΨδN

=δΨδNi

=∂Ψ∂t

= 0;(

δΨδhij

)∣∣∣∣|j

= 0

An interesting aspect of the WDW equation in (7.19) is that it has been obtainedwith a particular operator ordering, namely the supermetric Gijk is sandwichedbetween the two functional derivatives. This constrasts with the canonical ap-proach which is unable to specify a unique ordering (cf. [444] for references). Itshould be noted that different orderings can lead to different physical predictionsand hence the exact uncertainty approach is able to remove ambiguity in thisrespect. An analogous removal of ambiguity is obtained for quantum particleshaving a position dependent mass (cf. [449]) with the exact uncertainty approachspecifying, via (7.7), the unique sandwich ordering

(7.20) i∂ψ

∂t= −2

2∇ · 1

m∇ψ + V ψ

for the SE.

Summarizing now it follows that physical ensembles are described by a proba-bility density on configuration space (P), a corresponding conjugate quantity (S),and an ensemble Hamiltonian H[P, S]. The transition from classical ensembles toquantum ensembles then follows as a consequence of the addition of nonclassicalfluctuations, under the assumption that the fluctuation uncertainty is fully deter-mined by the configuration uncertainty. In contrast to the canonical approach theSE and WDW equations are derived, rather than postulated, and the probabilityconnection P = |ψ|2 is a simple consequence of the definition of ψ in terms of Pand S. Planck’s constant appears as a consequence of a derived universal scale forthe nonclassical momentum fluctuations instead of being an unexplained constantin the canonical approach. A (nonserious) limitation appear in that the momen-tum of a classical ensemble must contribute quadratically to the ensemble energyfor the exact uncertainty approach to go through whereas the canonical approachis indifferent to this. We refer to [444, 449] for more details and discussion.

REMARK 4.7.1. We extract here from the appendix to [449] for some nota-tion and constructions involving functional derivatives. One considers functionalsF [f ] with

(7.21) δF = F [f + δf ]− F [f ] =∫

dxδF

δfxδfx

Here f ∼ f(x) refers to fields with real or complex values. Thus the functionalderivative is a field density δF/δf having the value δF/δfx at position x. Forcurved spaces one would need a more elaborate notation, and volume element,etc. The choice F [f ] = fx′ in (7.21) yields δfx′/δfx = δ(x − x′) and if the fielddepends on a parameter t then writing δfx = fx(t + δt) − fx(t) one arrives at(♠) dF/dt = ∂tF +

∫dx(δF/δfx)∂tfx for the rate of change of F with respect

t. Functional integrals correspond to integration of functionals over the vectorspace of physical fields (or equivalence classes thereof) and the only property onerequires for the present discussion is the existence of a measure Df on this vectorspace which is translation invariant (i.e.

∫Df ≡

∫Df ′ for f ′ = f + h). This

property implies e.g.

(7.22)∫

DfδF

δf= 0 if

∫DfF [f ] <∞

This follows immediately via

(7.23) 0 =∫

Df(F [f + δf ]− F [f ]) =∫

dxδfx

(∫Df

δF

δfx

)In particular if F [f ] has a finite expectation value with respect to some probabilitydensity functional P [f ] then (7.22) gives an integration by parts formula

(7.24)∫

DfP (δF/δf) = −∫

Df(δP/δf)F

Moreover again via (7.22) the total probability∫

DfP is conserved for any prob-ability flow satisfying a continuity condition

(7.25)∂P

∂t+∫

dxδ

δfx[PVx] = 0

provided that the average flow rate < Vx > is finite. Next consider a functionalintegral of the form I[F ] =

∫Dfξ(F, δF/δf); then variation of I[F ] with respect

to F gives to first order

(7.26) ∆I = I[F + ∆F ]− I[F ] =

=∫

Df

(∂ξ/∂F )∆F +

∫dx[∂ξ/∂(δF/δfx)][δ(∆F )/δfx]

=

=∫

Df

(∂ξ/∂F )−

∫dx

δ

δfx[∂ξ/∂(δF/δfx)]

∆F+

+∫

dx

∫Df

δ

δfx[∂ξ/∂(δF/δfx)]∆F

One assumes here that Df is translation invariant and hence if the functionalintegral of the expression in curly brackets in the last term of (7.26) is finite theterm will vanish, yielding the result ∆I =

∫Df(∆I/∆F )∆F where ∆I/∆F =


∂F ξ −∫

dx(δ/δfx)[∂ξ/∂(δF/δfx)].

REMARK 4.7.2. Regarding the general HJ formulation of classical fieldtheory we go to Appendix B of [449]. Two classical fields f, g are canonicallyconjugate if there is a Hamiltonian functional H[f, g, t] such that

(7.27)∂f

∂t=

δH

δg;

∂g

∂t= −δH

δf

These equations follow from the action principle δA = 0 with A =∫

dt[−H +∫dxgx(∂fx/∂t))]. The rate of change of an arbitrary functional G[f, g, t] follows

from (7.27) and (♠) as

(7.28)dG

dt=

∂G

∂t+∫

dx

(δG

δfx

δH

δgx− δG

δgx

δH

δfx

)=

∂G

∂t+ G,H

A canonical transformation maps f, g,H to f ′, g′,H ′ such that the equations ofmotion for the latter retain the canonical form of (7.27). Equating the variationsof the corresponding actions A and A’ t zero it follows that physical trajectoriesmust satisfy

(7.29) −H +∫

dxgx(∂fx/∂t) = −H ′ +∫

dxg′x(∂f ′x/∂t) + (dF/dt)

for some generating functional F. Now any two of the fields f, g, f ′, g′ determinethe remaining two fields for a given canonical transformation; choosing f, g′ asindependent and defining the new generating functional G[f, g′, t] = F +

∫dxf ′

xg′xgives then via (♠)

(7.30) H ′ = H +∂G

∂t+∫

dx

[∂fx

∂t

(δG

δfx− gx

)+

∂g′x∂t

(δG

δg′x− f ′

x

)]The terms in round brackets therefore vanish identically yielding the generatingrealtions

(7.31) H ′ = H + ∂G/∂t; g = δG/δf ; f ′ = δG/δg′

A canonical transformation is thus completely specified by the associated generat-ing function. To obtain the HJ formulation of the equations of motion consider acanonical transformation to fields f ′, g′ which are time independent. From (7.27)one may choose the corresponding Hamiltonian H ′ = 0 without loss of general-ity and hence from (7.31) the momentum density and the associated generatingfunctional S are specified by

(7.32) g =δS

δf;

∂S

∂t+ H[f, δS/δf, t] = 0

the latter being the desired HJ equation. Solving this equation for S is equivalentto solving (7.27) for f and g.

Note that along a physical trajectory one has g′ = constant and hence from(♠) and (7.32)

(7.33)dS

dt=

∂S

∂t+∫

dxδS

δfx

∂fx

∂t= −H +

∫dxgx

∂fx

∂t=

dA

dt

Thus the HJ functional S is equal to the action functional A up to an additiveconstant. This relation underlies the connection between the derivation of theHJ equaiton from a particular type of canonical transformation as above and thederivation from a particular type of variation of the action. The HJ formulationhas the feature that once S is specified the momentum density is determined bythe relation g = δS/δf , i.e. it is a functional of f . Thus unlike the Hamiltonianformulation of (7.27) an ensemble of fields is specified by a probability densityfunctional P [f ], not by a phase space density functional ρ[f, g]. In either casethe equation of motion for the probability density corresponds to the conservationof probability, i.e. to a continuity equation as in (7.25). For example in theHamiltonian formulation the associated continuity equation for ρ[f, g] is

(7.34)∂ρ

∂t+∫

dx(δ/δfx)[ρ(∂fx/∂t)] + (δ/δgx)[ρ(∂gx/∂t)] = 0

which reduces to the Liuville equation ∂tρ = H, ρ via (7.27). Similarly in theHJ formulation the rate of change of f follows from (7.27) and (7.32) as

(7.35) Vx[f ] =∂fx

∂t=(

δH

δgx

)∣∣∣∣g=δS/δf

and hence the associated continuity equation for an ensemble of fields describedby P [f ] follows as in (7.25) to be

(7.36)∂P

∂t+∫

dxδ

δfx

[P

δH

δgx

∣∣∣∣g=δS/δf

]Everything generalizes naturally for multicomponent fields.

Given the background in Remarks 4.7.1 and 4.7.2 the development in [449]is worth sketching in connection with general bosonic field calculations. Thusone looks at the HJ formalism which provides a straightforward mechanism foradding momentum fluctuations to an ensemble of fields. First the equation ofmotion for an individual classical field is given by the HJ equation (•) ∂tS +H[f, δS/δf, t] = 0 where S[f ] denotes the HJ functional. The momentum densityassociated with the field f is g = δS/δf and hence S is called a momentumpotential. Next the description of an ensemble of such fields further requiresa probability density functional P [f ] whose equation of motion corresponds toconservation of probability, i.e. to the continuity equation (cf. (7.36))

(7.37)∂P

∂t+∑

a

∫dx

δ

δfax

(P

δH

δgax

∣∣∣∣g=δS/δf

)= 0

Equations (7.37) and (•) describe the motion of the ensemble completely via Pand S and this can be written in the Hamiltonian form

(7.38)∂P

∂t=

∆H

∆S;

∂S

∂t= −∆H

∆P; H[P, S, t] =< H >=

∫DfPH[f, (δS/δf), t]

(cf. Remark 4.7.1). The functional integral H in (7.38) will be referred to then asthe ensemble Hamiltonian and in analogy to (7.27) P and S may be regarded ascanonically conjugate functionals. Note from (7.38) that H typically corresponds

to the mean energy of the ensemble; moreover (7.38) follows from the action prin-ciple ∆A = 0 with action A =

∫dt[−H +

∫DfS(∂P/∂t)]. In the following one

specializes to ensembles for which the associated Hamiltonian is quadratic in themomentum field density, i.e. of the form

(7.39) H[f, g, t] =∑a,b

∫dxKab

x [f ]gaxgb

x + V [f ]

Here Kabx = Kba

x is a kinetic factor coupling components of the momentum densityand V [f ] is a potential energy functional. The corresponding ensemble Hamilton-ian is given via (7.38) and one notes that cross terms of the form ga

xgbx′ with x = x′

are not permitted in local field theories and hence are not considered here.

The approach of [449] to obtain a quantum ensemble of fields is now simply toadd nonclassical fluctuations to the momentum density with the magnitude of thefluctuations derermined by the uncertainty in the field. This leads to equationsof motion equivalent to those of a bosonic field with the advantage of a uniqueoperator ordering for the associated SE. Note here also the analogy with adding aquantum potential to the HJ equation in the Bohmian theory. Thus suppose nowthat δS/δf is in fact an average momentum density associated with the field f inthe sense that the true momentum density is given by

(7.40) g =δS

δf+ N

where N is a fluctuation field that vanishes on the average for any given f . Themeaning of S becomes then that of being an average momentum potential. Nospecific underlying model for N is assumed or necessary; one may in fact interpretthe source of the fluctuations as the field uncertainty itself. The main effect of thefluctuation field is to remove any deterministic connection between f and g. Sincethe momentum fluctuations my conceivably depend on the field f the average oversuch fluctuations for a given quantity A[f,N ] will be denoted by A[f ] and theaverage over fluctuations and the field by < A >. Thus N = 0 by assumptionand in general < A >=

∫DfP [f ]A[f ]. Assuming a quadratic dependence on

momentum as in (7.39) it follows that when the fluctuations are significant theclassical ensemble Hamiltonian H =< H > in (7.38) should be replaced by

(7.41) H ′ =< H[f, (δS/δf) + N, t] >= H +∑a,b

∫Df

∫dxPKab

x NaxN b

x =

=∑a,b

∫Df

∫dxPKab

x [(δS/δfax ) + Na

x ][(δS/δf bx) + N b

x]+ < V >

Thus the momentum fluctuations lead to an additional nonclassical term in theensemble Hamiltonian specified via the covariance matrix

(7.42) [Covx(N)]ab = NaxN b

x

This covariance matrix is uniquely determined (up to a multiplicative constant)by the following four assumptions:


(1) Causality: H ′ is an ensemble Hamiltonian for the canonical conjugatefunctionals P and S which yield causal equations of motion. Thus nohigher than first order functional derivatives can appear in the additionalterm in (7.41) which implies

Covx(N) = α

(P,

δP

δfx, S,

δS

δfx, fx, t

)for some symmetric matrix function α. Note the fourth assumption be-low removes the possibility of dependence here on auxillary fields andfunctionals.

(2) Independence: If the ensemble has two independent noninteractingsubensembles 1 and 2 with a factorisable probability density functionalP [f1, f2] = P1[f1]P2[f2] then any dependence of the corresponding N1, N2

on P only enters via P1, P2 in the form

Covx(N1)|P1P2 = Covx(N1)|P1 ; Covx(N2)|P1P2 = Covx(N2)|P2

This implies that the ensemble Hamiltonian H ′ in (7.41) is additive forindependent noninteracting ensembles (as is the corresponding actionA′).

(3) Invariance: The covariance matrix transforms correctly under linear canon-ical transormations of the field components. Thus f → Λ−1f, g → ΛT gis a canonical transformation for any invertible matrix Λ preserving thequadratic form of H in (7.39) and leaving the momentum potential Sinvariant (since δ/δf → ΛT δ/δf) and thus from (7.40) N → ΛT N soCovx(N) → ΛT covx(N)Λ for f → Λ−1f is required. For single com-ponent fields this reduces to a scaling relation for the variance of thefluctuations at each point x.

(4) Exact uncertainty: The uncertanty of the momentum density fluctua-tions, as characterized by the covariance matrix, is specified by the fielduncertainty and hence by the probability density functional P; henceCovx(N) cannot depend on S or explicitly on t.

One proves then in [449]

THEOREM 7.1. Under the above four assumptions one has

(7.43) NaxN b

x =C

P 2

δP

δfax

δP

δf bx

where C is a positive universal constant.

COROLLARY 7.1. The equations of motion corresponding to the ensembleHamiltonian H ′ can be expressed in the form

i∂Ψ∂t

= H

[f,−i

δ

δf, t

]Ψ = −2

⎛⎝∑a,b

∫dx

δ

δfax

Kabx [f ]

δ

δf bx

⎞⎠Ψ + V [f ]Ψ

where = 2√

C and Ψ =√

Pexp(iS/).

One notes that the corollary specifies a unique operator ordering for the func-tional derivative operators. The proofs are given below following [449] and this issubstantially different from (and stronger than) proofs of analogous theorems for

quantum particles in [447].

PROOF OF THEOREM AND COROLLARY:From the causality and exact uncertainty assumptions one has Covx(N) =

α(P, (δP/δfx), fx). Next to avoid issues of regularisation it is convenient to con-sider a position dependent canonical transformation fx → Λ−1

x fx such that A[Λ] =exp[

∫dxlog(|det(Λx)|)] < ∞. Then the probability density functional P and the

measure Df transform as P → AP and Df → A−1Df respectively (for conserva-tion of probability) and the invariance assumption requires

(7.44) α(AP,AΛTx u, Λ−1

x w) ≡ ΛTx α(P, u,w)Λx

where ua, wa denote respectively δP/δfax , fa

x for a given value of x. This musthold for A and Λx independently so choosing Λx to be the identity matrix at somepoint x, one has α(AP,Au,w) = α(P, u,w) for all A, which implies that α caninvolve P only via the combination v = u/P . Consequently

(7.45) α(ΛT v,Λ−1w) = ΛT α(v, w)Λ

This equation is linear and invariant under multiplication of α by any functionof the scalar J = vT w. Moreover one checks that if σ and τ are solutions thenso are στ−1σ and τσ−1τ . Choosing the two independent solutions σ = vvT andτ = (wwT )−1 it follows that the general solution has the form

(7.46) α(v, w) = β(J)vvT + γ(J)(wwT )−1

for arbitrary functions β and γ. Now for P = P1P2 one has v = (v1, v2) andw = (w1, w2) so the independence assumption reduces to the requirements

(7.47) β(J1 + J2)v1vT1 + γ(J1 + J2)(w1w

T1 )−1 = β1(J1)v1v

T1 + γ1(J1)(w1w

T1 )−1;

β(J1 + J2)v2vT2 + γ(J1 + J2)(w2w

T2 )−1 = β2(J2)v2v

T2 + γ2(J2)(w2w

T2 )−1

Thus β = β1 = β2 = C and γ = γ1 = γ2 = D for universal C and D yielding thegeneral form

(7.48) [covx(N)]ab = C1

P 2

δP

δfax

δP

δf bx

+ DW abx [f ]

where Wx[f ] denotes the inverse of the matrix with ab-coefficient faxf b

x. SinceWx[f ] is purely a functional of f it merely contributes a classical additive potentialterm to the ensemble Hamiltonian of (7.41); thus it has no nonclassical role andcan be absorbed directly into the classical potential < V >. In fact for fields withmore than one component this term is ill-defined and can discarded on physicalgrounds; consequently one can take D = 0 without loss of generality. Positivity ofC follows from positivity of the covariance matrix and the theorem is proved.

For the corollary one notes first that the equations of motion correspondingto the ensemble Hamiltonian H ′ follow via the theorem and (7.38) as first: Thecontinuity equation (7.37) as before (since the additional term does not depend


on S) and following (7.39) this takes the form

(7.49)∂P

∂t+ 2

∑a,b

∫dx

δ

δfax

(PKab

x

δS

δf bx

)= 0

and second: The modified HJ equation

(7.50)∂S

∂t= −∆H ′

∆P= −H[f, (δS/δf), t]− ∆(H ′ − H)

∆P

Calculating the last term via (7.43) and Remark 5.1 this simplifies to(7.51)

∂S

∂t+ H[f, (δS/δf), t]− 4CP−1/2

∑a,b

∫dx

(Kab

x

δ2P 1/2

δfax δf b

x

+δKab

x

δfax

δP 1/2

δf bx

)= 0

Now writing Ψ = P 1/2exp(iS/), multiplying each side of the equation in thecorollary by Ψ−1, and expanding, one obtains a complex equation for P and S.The imaginary part is just the continuity equation (7.49) and the real part is themodified HJ equation (7.50) provided C = 2/4.

EXAMPLE 7.1. This formulation is applied to gravitational fields for exam-ple to obtain a version of the WDW equation as in (7.18). Thus write (with somerepetition from before)

(7.52) ds2 = gµνdxµdxν = −(N2 −N ·N)dt2 + 2Nidxidt + hijdxidxj

as before and recall that the Einstein field equations follow from the Hamiltonianfunctional

(7.53) H[h, π,N,N) =∫

dxNHG[h, π]− 2∫

dxNiπij|j

where π = (πij) is the momentum density conjugate to h and |j denotes thecovariant 3-derivative. Further

(7.54) HG = (1/2)Gijk[h]πijπk − 2 3R[h](det(h))1/2

Here 3R is the scalar curvature corresponding to h and

(7.55) Gijk[h] = (hikhj + hihjk − hijhk)(det(h))−1/2

The Hamiltonian functional corresponds to the standard Lagrangian given viaL =

∫dx(−det(g))1/2R[g] where the momenta π0 and πi conjugate to N and Ni

vanish identically. However the lack of dependence of H on π0 and πi is consistentlymaintained only if the rates of change of these momenta also vanish, i.e. (noting(7.27), only if the constraints

(7.56)δH

δN= HG = 0;

δH

δNi= −2πij

|j = 0

are satisfied. Thus the dynamics of the field are independent of N and Ni so thatthese functions may be fixed arbitrarily. Moreover these constraints immediatelyyield H = 0 in (7.53) and hence the system is static with no explicit time de-pendence. It follows that in the HJ formulation of the equations of motion the


momentum potential S is independent of N, N, and t. Noting that π = δS/δh inthis formulation (7.56) yields the corresponding constraints

(7.57)δS

δN=

δS

δNi=

∂S

∂t= 0;

(δS

δhij

)∣∣∣∣|j

= 0

As shown in [759] a given functional F [h] is invariant under spatial coordinaetransformations if and only if (δF/δhij)||j = 0 and hence the fourth constraintin (7.57) is equivalent to the invariance of S under such transformations. Thisconstraint implies moreover that the second term in (7.53) may be dropped fromthe Hamiltonian yielding the reduced Hamiltonian

(7.58) HG[h, π,N ] =∫

dxNHG[h, π]

For an ensemble of classical gravitational fields the independence of the dynamicswith respect to (N, N, t) implies that members of the ensemble are distinguish-able only by their corresponding 3-metric h. Moreover it is natural to imposethe additional geometric requirement that the ensemble is invariant under spatialcoordinate transformations. Consequently one has constraints

(7.59)δP

δN=

δP

δNi=

∂P

∂t= 0;

(δP

δhij

)∣∣∣∣|j

= 0

The first two constraints imply that ensemble averages only involve integrationover hij .

Now in view of (7.54) the Hamiltonian HG in (7.58) has the quadratic formof (7.39) so the exact uncertainty approach is applicable and leads immediately tothe SE

(7.60) i∂Ψ∂t

=∫

dxNHG[h,−i(δ/δh)]Ψ

for a quantum ensemble of gravitational fields. One follows the guiding principleused before that all constraints imposed on the classical ensemble should be car-ried over to the corresponding constraints on the quantum ensemble. Thus from(7.57) and (7.59) one requires that P and S and hence Ψ in the equation of theCorollary above are independent of N, N, and t as well as invariant under spatialtransformations. Thus

(7.61)δΨδN

=δΨδNi

=∂Ψ∂t

= 0;(

δΨδhij

)∣∣∣∣|j

= 0

Applying the first and third of these constraints to (7.60) gives then, via (7.54),the reduced SE(7.62)

HG[h,−i(δ/δh)]Ψ = (−2/2)δ

δhijGijk[h]

δ

δhkΨ− 2 3R[h](det(h))1/2Ψ = 0

which is again the WDW equation.

CHAPTER 5

FLUCTUATIONS AND GEOMETRY

1. THE ZERO POINT FIELD

The zero point field (ZPF) arising from the quantum vacuum is still a con-tentious idea and we only make a few remarks following [18, 144, 145, 148, 183,251, 252, 253, 293, 310, 321, 438, 439, 440, 441, 442, 487, 488, 490, 606,650, 651, 753, 754, 791, 792, 793, 804, 813, 823, 825, 824, 955, 968, 969](see also [136, 377, 378, 379, 1006] for stochastic spacetime and gravity fluc-tuations). It is to be noted that a certain amount of research on ZPF is moti-vated (and funded) by the desire to extract energy from the vacuum for “spacetravel” and in this direction one is referred to publications of the CIPA (seehttp://www.calphysics.org/sci.html) where a number of papers discussed hereoriginate or are referenced. In a sense the quest here seems also to be an effort toreally understand the equation E = mc2. There is of course some firm physicalevidence for forces induced by quantum fluctuations via the Casimir effect for ex-ample and it is very stimulating to see so much speculation now in the literatureabout questions of mass, inertia, Zitterbewegung, and the vacuum. We extracthere first from [438] (cf. also [441]). No attempt is made here to be complete; thequantum vacuum is a relatively hot topic and there are many unsolved matters(for survey material see e.g. [655, 651, 753, 968, 969]). There are apparentlyat least two main views on the origin of the EM ZPF as embodied in QED andStochastic electrodynamics (SED). QED is “standard” physics and the argumentsgo as follows. The Heisenberg uncertainty principle sets a fundamental limit onthe precision wieh thich conjugate quantities are allowed to be determined. Thus∆x∆p ≥ /2 and ∆E∆t ≥ /2. It is standard to work via harmonic oscillators(see below) and there are two non-classical results for a quantized harmonic oscil-lator. First the energy levels are discrete; one can add energy but only in unitsof ν where ν is a frequency. The second stems from the fact that if an oscillatorwere able to come completely to rest ∆x would be zero and this would violate∆x∆p ≥ /2. Hence there is a minimum energy of ν/2 and the oscillator canonly take on values E = (n + 1/2)ν which can never be zero. The argumentis then made that the EM field is analogous to a mechanical harmonic oscillatorsince the fields E and B are modes of oscillating plane waves with minimum en-ergy ν/2. The density of modes between ν and ν + dν is given by the density ofstates function Nνdν = (8πν2/c3)dν. Each state has a minimum ν/2 of energyand thus the ZPF spectral density function is

(1.1) ρ(ν)dν = (8πν2/c3)(ν/2)dν

203

204 5. FLUCTUATIONS AND GEOMETRY

It is instructive to compare this with the blackbody radiation

(1.2) ρ(ν, T )dν =8πν2

c3

(ν

eν/kT − 1+

ν

2

)dν

If one takes away all the thermal energy (T → 0) what remains is the ZPF term.It is traditionally assumed in QM that the ZPF can be ignored or subtracted away.In SED is is assumed that the ZPF is as real as any other EM field and just camewith the universe. In this spirit the Heisenberg uncertainty relations for exampleare not a result of quantum laws but a consequence of ZPF. Philosophically onecould ask whether or not the universe is classical with ZPF (as envisioned at timesby Planck, Einstein, and Nerst and later by Nelson et al) or has two sets of laws,quantum and classical.

It was shown in the 1970’s that a Planck like component of the ZPF will arisein a uniformly accelerated coordinate system having constant proper accelerationa with what amounts to an effective temperature Ta = a/2πck (cf. [293]). Moreprecisely one says that an observer who accelerates in the conventional quantumvacuum of Minkowski space will perceive a bath of radiation, while an inertialobserver of course perceives nothing. This is a quantum phenomenon and thetemperature is negligible for most accelerations, becoming significant only in ex-tremely large gravitational fields. Thus for the case of no true external thermalradiation (T = 0), but including this acceleration effect Ta, equation (1.1) becomes

(1.3) ρ(ν, Ta)dν =8πν2

c3

[1 +

( a

2πcν

)2] [

ν

2+

ν

eν/kTa − 1

]dν

(the pseudo-Planckian component at the end is generally very small).

There have been (at least) two approaches demonstrating how a reaction forceproportional to acceleration (fr = −mZPa) arises out of properties of the ZPF.

force arising from the stochastically averaged magnetic component of the ZPF,namely < BZP >, as the basis of fr. The second, called RH after [823], considersonly the relativistic transformations of the ZPF itself to an accelerated frame,leading to a nonzero stochastically averaged Poynting vector (c/4π) < EZP ×BZP > which leads immediately to a nonzero EM ZPF momentum flux as viewedby an accelerating object. If the quarks and electrons in such an acceleratingobject scatter this asymmetric radiation an acceleration dependent reaction forcefr arises. In this context fr is the space part of a relativistic four vector sothat the resulting equation of motion is not simply the classical f = ma butrather the properly relativistic F = dP/dτ (which becomes exactly to f = mafor subrelativistic velocities). The expression for inertial mass in HRP for anindividual particle is mZP = Γzω2

c/2πc2 where Γz represents a damping constantfor Zitterbewegung oscillations (a free parameter) and ωc represents an assumedcutoff frequency for the ZPF spectrum (another free parameter). The expression

based on the first paperOne, called HRP and in [438], identified the Lorentz

1. THE ZERO POINT FIELD 205

for inertial mass in RH for an object with volume V0 is

(1.4) mi = mZP =(

V0

c2

∫η(ω)

ω3

2π2c3dω

)=

V0

c2

∫η(ω)ρZP dω

(note from (1.1) ω ∼ 2πν and mi here refers to inertial mass). Also from [440]recall that 4-momentum is defined as P = (E/c, p) = (γm0c, γm0v) where |P | =m0c and E = γm0c

2 (γ is a Lorentz contraction factor). Recall that the Comptonfrequency is given via νC = m0c

2 for a particle of rest mass m0 (note λC = /m0cso c = (/m0c)νC = λCνC ⇒ νC = c/λC has dimension T−1). Further λB = /pfor a particle of momentum p and here p ∼ m0γv so one expects

(1.5) λB = /m0γv = m0cλC/m0γv = (c/γv)λC

(cf. also [577]). In any event in [824] one argues that what appears as inertialmass in a local frame corresonds to gravitational mass mg and identifies mi in(1.4) with mg.

REMARK 5.1.1. We mention here a thoughtful analysis about the origin ofinertial mass etc. in [488] (second paper), which refers to the material developedabove in [438, 439, 823, 824]. It is worthwhile extracting in some detail as follows(cf. also [441, 487, 650, 753]). The purpose of the paper is stated to be that ofsuggesting qualifications in some claims that the classical equilibrium spectrum ofcharged matter is that of the classically conceived ZFF (cf. here [487] where oneintroduces an alternative classical ZPF with a different stochastic character whichreproduces the statistics of QED). It is pointed out that a classical massless chargecannot acquire mass from nothing as a result of immersion in any EM field andtherefore that the ZPF alone cannot provide a full explanation of intertial mass.Thus as background one mentions several works where classical representationsof the ZPF have been used to derive a variety of quantum results, e.g. the vander Waals binding (cf. [145]), the Casimir effect (cf. [650]), the Davies-Unruheffect (cf. [293]), the ground state behavior of the QM harmonc oscillator (cf.[487]), and the blackbody spectrum (cf. [145, 252]). In its role as the originatorof inertial mass the ZPF has been evisined as an external energizing influence fora classical particle whose mass is to be explained. In SED a free charged particleis deemed to obey the (relativistic version of the) Braffort-Marshall equation

(1.6) m0aµ −m0τ0

[daµ

dτ+

aλaλ

c2uµ

]= eFµλuλ

where τ0 = e2/6πε0m0c3 and f is the field tensor of the ZPF interpreted classically

(cf. [487]) for the correspondence between this and the vacuum state of the EMfield - indicated also later in this paper). If F is the ZPF field tensor operator then(1.6) is a relativistic generalization of the Heisenberg equation of motion for theQM position operator of a free charged particle, properly taking into account thevacuum sate of the quantized EM field (cf. [650]). From the standpoint of QED,once coupling to the EM field is switched on and radiation reaction admitted, theaction of the vacuum field is not an optional extra, but a necessary component ofthe fluctuation dissipation relation between atom and field. In the ZPF inertialmass studies the electrodynamics of the charge in its pre-mass condition has not


received attention, presumably on the grounds that the ZPF energization willquickly render the particle massive so that the intermediate state of masslessnessis on no import. However letting m0 → 0 one sees that Fµλuλ → 0 which demandsthat E·v → 0 (the massless particle moves orthogonal to E - cf. [488], first paper).It is concluded that if a charge is initially massless, there is no means by whichit can acquire inertial mass energy from an EM field, including the ZPF, andone must discount the possibility that the given ZPF can alone explain inertialmass of such a particle (this does not take into consideration however the possibleeffects of Dirac-Weyl geomety on mass creation - cf. Section 4.1). This is not todeny that mass may yet emerge from a process involving the ZPF or some otherEM field, only that it cannot be the whole story. Note here that in earlier works[441, 823] an internal structure is implied by the use of a mass-specific frequencydependent coupling constant between the charged particles and the ZPF so thereis no contradiction to the arguments above, namely to the statement that theclassical structureless charge particle cannot acquire mass solely as a result ofimmersion in the ZPF. One must also be concerned with a distinction betweeninertia as a reaction force and inertial mass as energy. For example one could askwhether the ZPF could be the cause of resistance to acceleration without it havingto be the cause of mass-energy. To address this consider the geometric form of themass action

(1.7) I = −m0c2

∫γ−1dt

which simultaneously gives both the mass acceleration fµ = m0aµ and the Noether

conserved quantity under time translations E = γm0c2 = m(v)c2 (i.e. the tradi-

tional mechanical energy (γ is the Lorenz factor). Thus the distinction betweenthe two qualities of mass appears to be one of epistemology. In some work (cf.[438]) the ZPF has been envisioned as an external energizing influence for an ex-plicitly declared local interal degree of freedom, intrinsic to the charged particlewhose mass is to be energized. Upon immersion in the ZPF this (Planck) oscilla-tor is energized and the energy so acquired is some of all of its observed inertialmass. Such a particle is not a structureless point in the usual classical sense andso does not suffer from an inability to acquire mass from the ZPF, provided theproposed components (sub-electron charges) already carry some inertia. There ismuch more interesting discussion which should be read. Note that one is excludinghere such matters as the (mythical) Higgs field as well as renormalization.

One of the first objections typically raised against the existence of a real ZPFis that the mass equivalent of the energy embodied in (1.1) would generate anenormous spacetime curvature that would shrink the universe to microscopic size(apparently refuted however via the principle of equivalence - cf. [438]). One notesthat all matter at the level of quarks and electrons is driven to oscillate (Zitter-bewegung) by the ZPF and every oscillating charge will generate its own minuteEM fields. Thus any particle wil experience the ZPF as modified ever so slightlyby the fields of adjacent particles - but that might be gravitation as a kind of longrange van der Waals force. A ZPF based theory of gravitation is however onlyexploratory at this point and there are disputes (see e.g. [183, 253, 310, 792]


- discussed in part later). In any event following [440, 441, 824] the preceedinganalysis leads to F = dP/dτ = (d/dτ)(γmic, p) for the relativistic force wheremi ∼ mZP ∼ mg. More generally via [823] (which improves [438]) plausiblearguments are given to show that the EM quantum vacuum makes a contribu-tion to the inertial mass mi in the sense that at least part of the inertial force ofopposition to acceleration, or inertia reaction force, springs from the EM quan-tum vacuum. Specifically, the properties of the EM vacuum as experienced ina Rindler constant acceleration frame were investigated, and the existence of anEM flux was discovered, called for convenience the Rindler flux (RF). The RF,and its relative, Unruh-Davies radiation, both stem from event horizon effects inaccelerating reference frames. The force of radiation pressure produced by the RFproves to be proportional to the acceleration of the reference fame, which leads tothe hypothesis that at least part of the inertia of an object should be due to theindividual and collective interaction of its quarks and electrons with the RF. Thisis called the quantum vacuum inertia hypothesis (QVIH). This is consistent withgeneral relativity (GR) and it answers a fundamental question left open withinGR, namely, whether there is a physical mechanism that generates the reactionforce known as weight when a specific nongeodesic motion is imposed on an object.Put another way, while geometrodynamics dictates the spacetime metric and thusspecifies geodesics, whether there is an identifiable mechanism for enforcing themotion of freely falling bodies along geodesic trajectories. The QVIH providessuch a mechanism since by assuming local Lorentz invariance (LLI) one can im-mediately show that the same RF arises due to curved spacetime geometry as foracceleration in flat spacetime. Thus the previously derived expression for the iner-tial mass contribution from the EM quantum vacuum field is exactly equal to thecorresonding contribution to the gravitational mass mg and the Newtonian weakequivalence principle mi = mg ensues. One also adopts the assumption of spaceand time uniformity (uniformity assumption (UA) stating that the laws of physicsare the same at any time or place within the universe. With such hypotheses onealso derives in [824] the Newtonian gravitational law f = −(GMm/r2)r while dis-claiming the success of such attempts in earlier work (cf. [183, 253, 439, 792]).We mention also some quite appropriate comments about the mystical Higgs fieldin [441] (cf. also [442]), the main point being perhaps that even if the Higgs existsit does not necessaarily explain inertial mass.

ZPF also plays the role of a Lorentz invariant EM component of a Dirac ether(cf. [439]). Thus Newton’s equation of motion in special relativity can be writ-ten as m(d2xµ/dτ2) = Fµ where four vectors are involved and Fµ represents anongravitational force. In general relativity this becomes

(1.8) m

(d2xµ

dτ2+ Γµ

νρ

dxν

dτ

dxρ

dτ

)= Fµ

The velocity dxµ/dτ is a time-like 4-vector and Fµ is a space-like 4-vector or-thogonal to the velocity. If Fµ = 0 in any coordinate frame it will be nonzero inall coordinate frames. The Γµ

νρ represent the gravitational force and can be setequal to zero by a coordinate transformation but Fµ cannot be transformed away.


Turning the arguments around the absolute nature of nongravitational accelera-tion is demonstrated by the manifestation of a force that cannot be transformedaway. Thus there is need for a special reference frame that is not perceptible onaccount of uniform motion but that is perceptible on account of non gravitationalacceleration and one proposes that the Lorentz invariant ZPF plays this role. Werefer to [144, 825] for background here.

REMARK 5.1.2. We mention for heuristic purposes some developmentsfollowing [442, 700] (cf. also [494] and Section 5.2 for possible comparison). Oneaddresses the quantum theoretic prediction that the Zitterbewegung of particlesoccurs at the speed of light, that particles exihbit spin, and that pair creation canoccur. Given motion at the speed of light the particles involved in Zitterbewegungwould seem to be massless (and in this respect we refer to [494]). A suitableequation for motion of a massless charge can be derived as a massless limit of theLorentz force equation maµ = (q/c)Fµνuν where q is charge and m will be goto zero; Fµν is the EM field tensor of impressed fields, including the ZPF. Theacceleration a and velocity u are four vectors and in terms of the usual three spacequanties a and v one can write

(1.9) m

[γ2aj + γ4v · avj

c

]=

q

cγ

[3∑1

F jkvk − F j0c

];

mγ4 v · ac

=q

cγ

[3∑1

F 0kvk − F 00c

]where γ = 1/

√1− (v2/c2). Combining leads to

(1.10) mγaj =q

c

[3∑1

(F jk − F 0k vj

c

)vk −

(F j0 − F 00 vj

c

)c

]Now let m → 0 and γ → ∞ with mγ remaining finite; thus one will havelimm→0,v→cmγ = m∗ and m∗ has the dimensions of mass but is not mass. Weneed further v = cn where n is a unit vector in the direction of the particle mo-tion. Acceleration can therefore only be due to changes in the direction of theform a = c(dn/dt) so that

(1.11) m∗cdnj

dt=

q

c

[3∑1

(F jk − F 0k vj

c

)vk −

(F j0 − F 00 vj

c

)c

]In terms of EM fields this is (dn/dt) = (q/m∗c)[n × B − (n · E)n + E]. Sincethe particle is moving at the speed of light it should see a universe Lorentz con-tracted to two transverse dimensions and can only be accelerated by forces fromthe side. When the impressed fields include the ZPF this motion can be regardedas Schrodinger’s Zitterbewegung. When a field above the vacuum is applied thecharge will be observed to drift in a preferred direction in its Zitterbewegung wan-dering. When viewed in a zoom picture one sees spin line orbital motion drivenby the ZPF. Now equation (1.11) does not exhibit inertia which seems in violationof relativity. However the ZPF fields have the vacuum energy density spectrum


of the form ρ(ω)dω = (ω3/2π2c3)dω where ω = 2πν (cf. (1.1)). The cubic fre-quency dependence endows the spectrum with Lorentz invariance and all inertialframes see an isotropic ZPF. A Lorentz transformation will cause a Doppler shiftof each frequency component but an equal amount of energy is shifted into and outof each frequency bin. When there are no fields above the vacuum in an inertialframe an observer in that frame should expect to see a zero-mean random walkdue to the isotropic ZPF. Thus in our example an observer in a frame comovingwith the average motion of the charge just before the driving field is switched offshould expect to see continued zero-mean Zitterbewegung in his frame whereas(1.11) produces zero-mean motion in whatever frame the calculation is performed.To have a consistent theory Lorentz covariance must be restored. Thus assume(1.11) holds in the spacetime of the particle and describes a null geodesic. Thecurvature is defined by the EM fields in the particles history and since the par-ticle is massless and moving at the speed of light one cannot use proper time asthe affine parameter of the geodesic (since proper time intervals vanish for nullgeodesics). However normal time serves as well for time parameter and (1.11) isreplaced with

(1.12)dpµ

dt+

1m∗

Γµνρp

νpρ = 0; pµ = m∗cnµ; nµ = (n0,n)

One equates the connection terms with the Lorentz force terms of maµ = (q/c)Fµνuν ,i.e.

(1.13) Γµνρp

νpρ = −(q/c)Fµν pν

and these equations can be solved for the metric of the particle’s spacetime (al-though not uniquely - there is apparently a class of metrics). Further constraintsare needed to select a particular solution and in particular the geodesic should bea null curve as expected for a massless object. Using (1.13) the geodesic equationin (1.12) is claimed to be

(1.14)dnj

dt=

q

m∗c

[F j

ν nν − F oν nν nj

n0

]+

nj

n0

dn0

dt;

dm∗dt

=q

cF 0

ν

nν

n0− m∗

n0

dn0

dt

In general n0 does not retain a value of 1 but should change in a way that pre-serves the null curve property gµνpµpν = 0. We note that the second equationin (1.14) is an equation for the parameter m∗, which is therefore not a constantbut rather varies in response to applied forces. The effect is to introduce timedilation (or Doppler shifting) in the EM 4-vector analogous to the gravitationalredshift of GR. Also note that inertia is not assumed here by the requirementthat the particle travel on a null geodesic; the particle is not restricted but itsmotion is used to define the spacetime and metric that the particle sees. It is onlyafter a transformation to Minkowski spacetime that inertial behavior appears. Tofind solutions of (1.13) consider an infinitesimal region around the charge andrequire gµν = ηµν + gµν,ρdxρ where ηµν is the flat spacetime metric (signature(−1, 1, 1, 1)). The Christoffel symbols are then calculated in terms of the deriv-atives gµν,ρ and substituted into (1.13). A simple local solution is written downwith some interesting properties for which we refer to [442, 700].


1.1. REMARKS ON THE AETHER AND VACUUM. Regardingmassless particles and Maxwell’s equations we refer here to [9, 19, 29, 152, 233,331, 405, 442, 543, 666, 700, 863, 864, 920] and consider first [405]. Thus theDirac equation is derived from the relativistic condition (E2−c2p2−m2c4)I4ψ = 0where I4 is a 4×4 unit matrix and ψ is a fourcomponent (bispinor) wave function.This can be decomposed via

(1.15)[EI4 +

(mc2I2 cp · σcp · σ −mc2I2

)]×[EI4 −

(mc2I2 cp · σcp · σ −mc2I2

)]ψ = 0;

σx =(

0 11 0

); σy =

(0 −ii 0

); σz =

(1 00 −1

)and I2 is a 2×2 unit matrix. The two component neutrino equation can be derivedfrom the decomposition

(1.16) (E2 − c2p2)I2ψ = [EI2 − cp · σ][EI2 + cp · σ[ψ = 0

where ψ is a two component spinor wavefunction. The photon equation can bederived from the decomposition(1.17)(

E2

c2− p2

)I3 =

(E

cI3 − p · S

)(E

cI3 + p · S

)−

⎛⎝ p2x pxpy pxpz

pypx p2y pypx

pxpz pxpy p2z

⎞⎠ = 0;

Sx =

⎛⎝ 0 0 00 0 −i0 i 0

⎞⎠ ; Sy =

⎛⎝ 0 0 i0 0 0−i 0 0

⎞⎠ ; Sz =

⎛⎝ 0 −i 0i 0 00 0 0

⎞⎠and I3 is a 3 unit matrix. Further

(1.18) [Sx, Sy] = iSz, [Sz, Sx] = iSy, [Sy, Sz] = iSx, S2 = 2I3

One notes that the last matrix in (1.17) (first line) can be written as (px py pz)T ·(px py pz). This leads to the photon equation in the form

(1.19)(

E2

c2− p2

)ψ =

(E

cI3 − p · S

)(E

cI3 + p · S

)ψ −

⎛⎝ px

py

pz

⎞⎠ (p · ψ) = 0

where ψ is a 3-component column wave function. (1.19) will be satisfied if

(1.20)(

E

cI3 + p · S

)ψ = 0; p · ψ = 0

For real energies and momenta conjugation leads to

(1.21)(

E

cI3 − p · S

)ψ∗ = 0; p · ψ∗ = 0


The only difference here is that (1.20) is the negative helicity equation while (1.21)represents positive helicity. Now in (1.20) make the substitutions E ∼ i∂t andp ∼ −i∇ with ψE− iB. Then (p · S)ψ = ∇× ψ and this leads to

(1.22)i

c∂tψ = −∇× ψ; −i∇ · ψ = 0

Cancelling (!) one obtains then

(1.23) ∇× (E− iB) = − i

c∂t(E− iB); ∇ · (E− iB) = 0

If the electric and magnetic fields are real one obtains the Maxwell equations

(1.24) ∇×E = −1c∂tB; ∇×B =

1c∂tE; ∇ ·E = ∇ ·B = 0

The Planck constant does not appear since it cancelled out! In any event thisshows the QM nature of the Maxwell equations and hence of EM fields.

REMARK 5.1.3. One notes here a similar formula arising in [9] going backto work of Majorana, Weinberg, et al (see [9] for references). The idea thereinvolves the matrices S of (1.17) written as s1, s2, s3 with

(1.25) i∂E∂t

=1i(s · ∇)iB; i

∂(iB)∂t

=1i(s · ∇)E; E =

⎛⎝ E1

E2

E3

⎞⎠ ; B =

⎛⎝ B1

B2

B3

⎞⎠corresponding to the first two equations in (1.24). Then a fermion-like formulationis created via 6× 6 matrices

(1.26) Γ0 =(

I 00 −I

); Γ =

(0 s−s 0

); Γ5 =

(0 II 0

)For S =

(0 ss 0

)(1.25) becomes

(1.27) i∂tψ = (1/i)(S · ∇)ψ ⇒ (∂t + S · ∇)ψ = 0

Equation (1.27) resembles the massless Dirac equation and one can consider ψ as a(quantum) wave function for the photon of type ψ = (E iB)T with ψ = (E† iB†)where ψ = ψ†Γ0 is an analogue of Hermitian conjugate. Note ψ†ψ = E2 + B2

(where E† = E and B† = B) corresponds to the local mean number of photons.The construction of ψ thus mimics a Dirac spinor and one can write

(1.28) iΓ0∂ − tψ =1i(Γ0S · ∇)ψ =

1i(Γ · ∇)ψ ⇒ (Γ0∂t + Γ · ∇)ψ = 0

or more compactly Γµ∂µψ = 0 (although this is not manifestly covariant). Onegoes on to construct a Lagrangian with a duality transformation and we refer to[9] for more details.

REMARK 5.1.4. Going to [331] one finds some generalizations of [405] inthe form

(1.29) ∇×E = −1c∂tB +∇(χ); ∇×B =

1c∂tE +∇(χ);


∇ ·E = −1c∂t(χ); ∇ ·B =

1c∂t(χ)

and some further analysis of spin situations.. If one assumes no monopoles it maybe suggested that χ(x)is a real field and its derivatives play the role of charge andcurrent densities; there is also some flexibility here in interpretation.

1.2. A VERSION OF THE DIRAC AETHER. We go here to [215,216] (cf. also [302, 499, 500, 727]). Dirac adopted the idea of using spuriousdegrees of freedom associated to the gauge potential to describe the electron viaa gauge condition A2 = k2. In fact in [302] this was introduced as a gauge fixingterm in the Lagrangian L = −(1/4)F 2 + (λ/2)(A2− k2) leading to ∂νFµν = Jµ ≡λAµ. Here the gauge condition doesn’t intend to eliminate spurious degrees offreedom but rather acquires a physical meaning as the condition allowing the rightdescription of the physics without having to introduce extra fields. From this Diracargued that it would be possible to consider an aether provided that one interpretsits four velocity v as a quantity subjected to uncertainty conditions. Admitting theaether velocity as defining a point in a hyperboloid with equation v2

0−v2 = 1 withv0 > 0 it could be related to the gauge potential (satisfying A2 = k2) via (1/k)Aµ =vµ. Then v (the aether velocity) would be the velocity with which an electric chargewould flow if placed in the aether. The model in [215] is a continuation of [727]and it is implicit in the present formulation that there is an inertial frame (theaether frame) in which the aether is at rest; one writes (1, 0, 0, 0) (∼ vµ

aether) forthe aether frame. Then an action S =

∫dx(−(1/4)F 2 + σvαFαµAµ) is proposed

with v being the aether’s velocity relative to a generic observer (inertial or not).For inertial observers vµ = Λµ

νvνaether = Λµ

0 is still constant and one obtains fromthe equations of motion a current Jµ = −σvµ∂ ·A + σvν∂µAν that is understoodas being induced in the aether by the presence of the EM field. Therefore it definesa polarization tensor Mαβ from which one obtains the vectors of polarization andmagnetization of the medium. In the aether reference frame this allows one todefine the electric displacement vector as D = E+σA while H = B; the resultingequations are similar in form with the macroscopic Maxwell equations in a medium,in agreement with [537, 959] but with the difference that D = εE. Hence theaether cannot be thought of as an isotropic medium. Moreover in a generic refernceframe moving relative to the aether D and H will depend on v. There is moresummary material in [215] but we proceed directly here to the equations.

For flat spacetime one takes for ηµν = diag(+,−,−,−)

(1.30) xµ = (x0, xi) = (t,x); xµ = (x0, xi) = (t,−x); Aµ = (A0, Ai) = (φ,−A);

∂µ = (∂0, ∂i) = (∂t,−∇); ∂µ = (∂0, ∂i) = (∂t,∇); Aµ = (A0, Ai) = (φ,A);

F0i = Ei; Fij = −εijkBk; F 0i = −F0i = −Ei; F ij = Fij = −εijkBk

Take now

(1.31) S =∫

dx(−(1/4)F 2 + J ·A);

Jµ = σvαFαµ; Fµν = ∂µAν − ∂νAµ


The constant σ is associated to the aether conductivity and J will be conserved(we drop the bold face here); J · A defines an interaction of the gauge field withitself. Then the equation of motion for Aµ is

(1.32) ∂νF νµ + σvµ∂ ·A− σvν∂µAν = 0

and this assumes the form of the Maxwell equations in the presence of a source∂νF νµ ≡ Jµ provided that one identifies Jµ = −σvµ∂ · A + σvν∂µAν with aconserved 4-current. This is the same as Dirac in which the term jµ = λAµ isinterpreted as a 4-current but here this arises from the interaction term J · Ainstead of via a gauge fixing term (1/2)λ(A2 − k2) in the action. Now taking thedivergence of ∂νF νµ = Jµ one gets

(1.33) 0 = ∂µJµ = σvµ(Aµ − ∂µ∂ ·A) = σvµ∂νF νµ = σvµJµ = σ2(−v2∂ ·A+

+vαvβ∂αAβ); ∂ ·A = (vαvβ/v2)∂αAβ

This constraint is a new feature of this model; its origin is independent of anylocal symmetry of the action.

For global gauge invariance one considers an invariance defined via the equa-tion Aµ → A′

µ = Aµ + λvµ (where ∂µλ = 0); this is associated to the Noethercurrent

(1.34) Θµ = Fµνvν − σvµA · v + σv2Aµ ≡ Θµνvν

where Θµν = Fµν − σvµAν + σvνAµ. Later this Θµν will be interpreted as Hµν

which in the aether frame becomes Hµν = (D,H). In a system at rest relativeto the aether one has Θµ = (0,E + σA) and (1.33) implies ∇ · A = 0. Thenthe conservation equation for Θµ gives ∇ · E = 0. In this model there is anotherconserved current

(1.35) Jµ = −σvµ∂ ·A + σv · ∂Aµ

where J = J − J so that conservation of J will follow immediately. Equivalentlyone can think of Jν as originating from the divergence of Θµν , i.e. ∂µΘµν =−Jν . In the classical formulation of electrodynamics in conducting media thenonhomogeneous Maxwell equations are written covariantly as ∂µHµν = −jν

ext

with Hµν having D and H as its components. Here Θµν above generalizes Hµν

and ∂µΘµν = −Jν corresponds to ∂µH = −jνext. This allows one to interpret Jµ as

the corresponding 4-current in much the same way as Dirac interpreted jµ = λAµ

as a 4-current in [302]. The interaction term J ·A then parallels the same term inthe usual electrodynamics. The global symmetry above is a new feature with nocounterpart in the Maxwell formulation. It is also possible to add a mass term toS above that preserves this global symmetry; in fact the action

(1.36) S =∫

dx(−(1/4)F 2 + J ·A− (1/2)σ2Aµ(v2gµν − vµvν)Aν)

is invariant. From this one obtains the equation

(1.37) ∂νF νµ ≡ Jµ = −σvµ∂ ·A + σvν∂µAν + σ2(v2gµν − vµvν)Aν ∼

∼ [gµν(− σ2v2) = ∂µ∂ν + σ(vµ∂ν − vν∂µ) + σ2vµvν ]Aν = 0


Here the conservation of the current J that follows from (1.37) doesn’t produceany constraint on A; further the conserved current in (1.34) is associated to theglobal symmetry. Finally the local gauge invariance depends on a parameter θ(x)and has the usual form Aµ → A′

µ = Aµ + ∂µθ and (1.33) adds some new featuresto the analysis. In fact let A′ and A be two fields related by A′

µ = Aµ +∂µθ. Sinceboth field configurations should obey (1.33) one must have

(1.38) ∂ ·A′ = (1/v2)(v · ∂)(v ·A′) ⇐⇒ θ = (vαvβ/v2)∂α∂βθ

Now let ∂ · A = 0 and choose θ so that it ensures ∂ · A′ = 0; one should thenhave θ satisfying θ = −∂ · A. This last condition together with the constraints(1.33) and (1.38) give then ∂α∂βθ = −(1/2)(∂αAβ + ∂βAα) which represents astronger restriction than that shown in θ = −∂ ·A. Equivalently one can obtainthis stronger restriction directly from 0 = ∂ · A′ = (vαvβ/v2)∂α(Aβ + ∂βθ) using(1.33).

Now one considers this model in the aether reference frame where v = (1, 0, 0, 0)and it is supposed that the aether is a medium without any given density of chargeor current. From ∂νF νµ = Jµ one has

(1.39) ∇ ·E = −σ∇ ·A; ∇×B =∂E∂t

+ σ∂A∂t

+ σE

to which is added the homogeneous equations ∇ · B = 0 and ∇ × E = −∂tB.Since ∇ · E = 0 we see that (1.39) introduces a new feature for the physicalvacuum (cf. also [597]). Essentially a divergenceless equation for E signals that thevacuum is not merely an empty space but it is also capable of becoming electricallypolarized. The presence of additional terms depending on the potential vector in(1.39) indicates the response of the medium to the presence of the fields (E,B),which resembles the phenomena of polarization and magnetization of a medium.Therefore one rewrites (1.39) as ∇ ·D = 0 with D = E + σA +∇ ×K. At thispoint K is an arbitrary vector that can be thought of as playing the role of a gaugeparameter. Now rewrite (1.39) with an assignment for K as

(1.40) ∇×B = ∂tD + σE− ∂t∇×K with ∂t∇×K = σE

Then using ∇ ·B = 0 and ∇×E = −∂tB one obtains ∂t∇× (σA +∇×K) = 0.The vector K can be still further restricted so that σA+∇×K = 0 and this givesE = D and B = H with the Maxwell equations in free space, namely

(1.41) ∇ ·E = 0; ∇×B = ∂tE; ∇ ·B = 0; ∇×E = −∂tB

Conditions (1.40) with σA + ∇ ×K = 0 can be interpreted as originating fromthe imposition of the temporal gauge and together they imply ∇A0 = 0 which isnaturally satisfied by putting A0 = 0. It is possible to give another description forthe present electrodynamics without using K. Thus from (1.39) one can simplyidentify D = E + σA and B = H, leading to

(1.42) ∇ ·D = 0; ∇×B = ∂tD + σE; ∇ ·B = 0; ∇×E = −∂tB

and this coincides with the aether equations of [537, 959]. In the identificationD = E + σA and B = H the aether behaves like a medium that responds to thepresence of the electric field by creating a polarization P = σA. One also has a


curreent J = σE which is in agreement with the supposition of the aether beinga medium with conductivity σ. According to Schwinger’s idea of a structurelessvacuum an EM field disturbs the vacuum affecting its properties of homogeneityand isotropy. This is exactly the situation obtained in the present model wherethe presence of an EM field in a vacuum with conductivity σ produces a responseof the medium (D = εE) that signals its nonisotropy.

Thus for a flat spacetime and a reference frame at rest relative to the aetherone has D = E + σA and B = H. In the case of a curved spacetime and anoninertial reference frame moving relative to the aether one will have a morecomplicated relation between H and F. Indeed in a medium that is at rest in anyreference fame with a metric gαβ one knows from [967] that the relation betweenH and F has the form

(1.43)√−gHαβ =

√−ggαγgβκSµ

γ SνκFµν

where Sαβ characterizes the EM properties of the medium. It is convenient to

rewrite (1.43) as

(1.44)√−gHαβ =

√−ggαγgβκSµ

γκAµ;

Sµγκ = (Sν

γSµκ − Sµ

γ Sνκ)∂ν

As an application of (1.43) it was shown in [967] that for the vacuum (consideredfrom an inertial reference frame) the tensor Sα

β has the form Sαβ = δα

β and the ma-terial equations become

√−gHαβ =√−ggαµgβνFµν which reproduces the usual

equations of free electrodynamics in a curved background (cf. [793]). Also in thecase of a linear isotropic medium that is at rest in an inertial reference frame thetensor Sα

β is given by S00 = ε

√µ, S1

1 = S22 = S3

3 = 1/√

µ and one obtains the usualrelations D = εE and H = (1/µ)B. In the present model in order to define Hαβ

for a generic reference frame in a curved background and t find a suitable materialrelation of the type (1.44) one should first follow the preceeding approach whichallows one to define Hµν directly from the equation of motion for Aµ. Explicitlytake the equation of motion in a curved background to be

(1.45) ∂ν(√−gFµν − σ

√−gvνAµ + σ

√−gvµAν) = Jµ = −σ

√−gvνF νµ

Then one defines

(1.46)√−gHαβ =

√−g(Fαβ − σvαAβ + σvβAα)

This is equivaent to the introduction of an antisymmetric polarization tensor Mαβ

(cf. [967]) of the form

(1.47)√−gHαβ =

√−g(Fαβ + Mαβ) ⇐⇒

√−gHαβ =

√−ggαν(Fνµ + Mνµ)

provided we identify

(1.48) Mαβ = −σvαAβ + σvβAα ⇐⇒ Mαβ = −σvαAβ + σvβAα;

Fµν = gµαgνβFαβ , Fµν = ∂µAν − ∂νAµ; Mµν = gµαgνβMαβ

Finally in order to obtain the material equations one extends (1.44) by allowingSµ

αβ to be a generic operator not restricted as in (1.44) but given by Sµαβ = δµ

β (∂α−σvα) − δµ

α(∂β − σvβ). Here the tensor Sµαβ may contain not only EM properties


of the medium (as in the case of (1.44)) but also information about the referenceframe (implicit in the 4-velocity vα). Adopting the convention Di =

√−gHi0 andHi = −(1/2)εijk

√−gHjk (cf. [967]) and considering a flat spacetime one obtains(in the aether frame) the relations D = E + σA and B = H.

1.3. MASSLESS PARTICLES. We review again some classical and quan-tum features of EM with some repetition of earlier material. First, following [650],recall the Maxwell equations

(1.49) ∇ · E = 0; ∇ ·B = 0; ∇× E = −1cBt; ∇×B =

1cEt

(bold face is omitted). Recall now the vector potential A eners via B = ∇ × Aand from the third equation in (1.49) E = −(1/c)At − ∇φ where φ is the scalarpotential. Hence ∇2A− (1/c2)Att = 0 in the Coulomb gauge defined via ∇·A = 0and in the absence of sources φ = 0. Separation of variables gives monochromaticsolutions

(1.50) A(x, t) = α(t)A0(x) + α∗(t)A∗0(x) = α(0)e−iωtA0(x) + α∗(0)eiωtA∗

0(x)

where (F2) ∇2A0(x) + k2A0(x) = 0 (k = ω/c) and α = −ω2α. Consequently

(1.51) E(x, t) = −1c[α(t)A0(r) + α∗(t)A∗

0(x);

B(x, t) = α(t)∇×A0(x) + α∗(t)∇×A∗0(x)

and a calculation gives the EM energy as

(1.52) HF = (1/8π)∫

d3x(E2 + B2) = (k2/2π)|α(t)|2

where A0 is normalized via∫

d3x|A0(x)|2 = 1 Now defining

(1.53) q(t) =i

c√

4π[α(t)− α∗(t)]; p(t) =

k√4π

[α(t) + α∗(t)]

gives HF = (1/2)(p2 + ω2q2) so the field mode of frequency ω is mathematicallyequivalent to a harmonic oscillator of frequency ω. Note q and p are canonicallyconjugate since q = p and p = −ω2q corresponds to Hamiltonian equations withHamiltonian HF .

REMARK 5.1.5. We assume familiarity with harmonic oscillator calcula-tions. For H = (p2/2m) + (1/2)mω2q2 one has q = (i)−1[q, H] = p/m andp = (i)−1[p,H] = −mω2q. Then defining

(1.54) a =1√

2mω(p− iq); a† =

1√2mω

(p + imωq) ≡

≡ q = i

√

2mω(a− a†); p =

√mω

2(a + a†)

which yields [q, p] = i and [a, a†] = 1 with H = (1/2)ω(aa† + a†a) = ω(a†a +(1/2)). For N = a†a one has eigenkets N |n >= n|n > and a|n >=

√n|n − 1 >

with a†|n >=√

n + 1|n + 1 >; further En = [n + (1/2)]ω.

One can now replace (1.50) by

(1.55) A(x, t) =(

2πc2

ω

)1/2

[a(t)A0(x) + a†(t)A∗0(x)];

E(x, t) = i(2πω)1/2[a(t)A0(x)− a†(t)A∗0(x)];

B(x, t) =(

2πc2

ω

)1/2

[a(t)∇×A0(x) + a†(t)A∗0(x)]

and HF becomes HF = ω[a†a+(1/2)]. Now the vacuum state |0 > has no photonsbut has an energy (1/2)ω and QM thus predicts the ZPF. In all stationary states|n > one has < E(x, t) >=< B(x, t) >= 0 since < n|a|n >= 0. This means thatE and B fluctuate with zero mean in the state |n > even though the state has adefinite nonfluctuating energy [n + (1/2)]ω. A computation also gives

(1.56) < E2(x, t)[n + (1/2)]4πω|A0(x)|2 = 4πω|A0(x)|2n+ < E2(x) >0

The factor n is the number of photons and |A0(x)|2 gives the same spatial intensityas in the classical theory. Normal ordering is used of course in QM (a† to the leftof a) and this eliminates contributions of ZPF to various calculations. Howeverone does not eliminate the ZPF by dropping its energy from the Hamiltonian. Inthe vacuum state E and B do not have definite values (they fluctuate about azero mean value) and one arrives at energy densities etc. as before. An atom isoften considered to be “dressed” by emmision and reabsorption of virutal photonsfrom the vacuum which itself has an infinite energy. Some thermal aspects werementioned before (cf. (1.2) for example) and this is enormously important (in thisdirection there is much information in [650, 950]).

REMARK 5.1.6. We refer next to the first paper in [488] on massless clas-sical electrodynamics for some interesting calculations (some details are omittedhere). One considers a bare charge, free of self action, compensating forces, andradiation reaction (cf. [488] for a long discussion on all this). Use the conventionuava = u0v0 − u · v and Heavyside-Lorentz units with c = 1 in general. Withthe fields given the Euler equation for the (massless) lone particle degree of free-dom is simply that the Lorentz force on the particle in question must vanish, i.e.F νµuµ = 0 where F is the EM field stregth tensor and the fields E, B are to beevaluated along the trajectory. In 3 + 1 form, omitting particle labels, this is

(1.57) (dt(λ)/dλ)E(x(λ), t(λ)) + (dx(λ)/dλ)×B(x(λ), t(λ)) = 0

In order for F νµuµ = 0 to have a solution the determinant of F must vanish whichgives

(1.58) S(x(λ)) ≡ E(x(λ)) ·B(x(λ)) = 0

This imposes a constraint on the fields along the trajectory which can be inter-preted as the condition that the Lorentz force on a particle must vanish (recognizedas the constraint on the fields such that there exist a frame in which the electricfield is zero). Calculation leads to

(1.59) v(x, t) =dx/dλ

dt/dλ=

E ×∇S −BSt

B · ∇S


as the ordinary velocity of the trajectory passing through (t(λ), x(λ)). The rightside is an arbitrary function of (x, t) decided by the fields and in general (1.59)will not admit a solution of the form x = f(t) since the solution trajectory maybe nonmonotonic in time. Various situations are discussed including the particlesadvanced and retarded fields. In particular the particle does not respond to forcein the traditional sense of Newton’s second law. Its motion is precisely that whichcauses it to feel no force. Yet its motion is uniquely prescribed by E and B, whichdecide the particle trajectory (given some initial condition) just as the Lorentzforce determines the motion of a massive particle. The important difference isthat traditionally the fields determine acceleration whereas here they determinethe velocity. The massless particle discussed cannot be a relative of the neutrinoand it does not seem to be a traditional classical object in need of quantization.

1.4. EINSTEIN AETHER WAVES. We go now to [510] where the vi-olation of Lorentz invariance by quantum gravity effects is examined (cf. also[28, 51, 337, 338, 380, 459, 518, 642]). In a nongravitational setting it suf-fices to specify fixed background fields violating Lorentz symmetry in order toformulate the Lorentz violating (LV) matter dynamics. However this would breakgeneral covariance, which is not an option, so one promotes the LV backgroundfields to dynamical fields, governed by a generally covariant action. Virtually anyconfiguration of matter fields breaks Lorentz invariance but here the LV fieldscontemplated are constrained dynamically or kinematically not to vanish, so thatevery relevant field configuration violates local Lorentz symmetry everywhere, evenin the “vacuum”. If the Lorentz violation preserves a 3-dimensional rotation sub-group then the background field must be a timelike vector and one considers herethe case where the LV field is a unit timelike vector ua which can be viewed as theminimal structure required to determine a local preferred rest frame. One optsto call this field the “aether” as it is ubiquitous and determines a local preferredrest frame. Kinetic terms in the action couple the ether directly to the spacetimemetric in addition to any couplings that might be present between the aether andthe matter fields. This system of the metric coupled to the aether will be referredto as Einstein-aether theory. In [401] an essentially equivalent theory appearsbased on a tetrad formalism (cf. also [510, 642] for various special cases).

In the spirit of effective field theory consider a derivative expansion of theaction for the metric gab and aether ua. The most general action that is diffeo-morphism invariant and quadratic in derivatives is

(1.60)S = 1

16πG

∫d4x√−g(−R + Lu − λ(jaua − 1)); Lu = −Kab

mn∇aum∇bun;

Kabmn = c1g

abgmn + c2δamδb

n + c3δanδb

m + c4uaubgmn

Here R is the Ricci scalar and λ is a Lagrange multiplier enforcing the unit con-straint. The signature is (+,−,−,−) and c = 1 (cf. [972] for other notation).The possible term Rabu

aub is proportional to the difference of the c2 and c3 termsvia integration by parts and hence has been omitted. Also any matter coupling


is omitted since one wants to concentrate on the metric-aether sector in vacuum.Varying the action with respect t ua, gab, and λ yields the field equations

(1.61) ∇aJam = c4ua∇mua = λum; Gab = Tab; gabu

aub = 1

Here one has Jam = Kab

mn∇bun and um = ua∇aum and the aether stress tensor

is

(1.62) Tab = ∇m(J m(a ub) − Jm

(aub) − J(ab)um)+

+c1[(∇mua)(∇mub)− (∇aum)(∇bum)] + c4uaub+

+[un(∇mJmn)− c4u2]uaub −

12gabLu

Here the constraint has been used to eliminate the term that arises from varying√−g in the constraint term in (1.60) and in the last line λ has been eliminatedusing the aether field equations.

Now the first step in finding the wave modes is to linearize the field equaitonsabout the flat Minkowski background ηab and the constant unit vector ua. Theexpanded fields are then

(1.63) gab = ηab + γab; ua = ua + va

The Lagrange multiplier λ vanishes in the background so we use the same notationfor its linearized version. Indices will be raised and lowered now with ηab andone adopts Minkowski coordinates (x0, xi) aligned with ua; i.e. for which ηab =diag(1,−1,−1,−1) and ua = (1, 0, 0, 0). Keeping only the first order terms in va

and γab the field equations become

(1.64) ∂aJ (1)am = λum; G

(1)ab = T

(1)ab ; v0 +

12γ00 = 0

where the superscript (1) denotes the first order part. The linearized Einsteintensor is

(1.65) G(1)ab = −1

2γab −

12γ,ab + γ m

m(a,b) +12ηab(γ − γ mn

mn, )

where γ mm is the trace, while the linearized aether stress tensor is

(1.66) T(1)ab = ∂m[J (1)m

(a ub) − J(1)m(a ub) − J

(1)(ab)u

m] + [un(∂mJ (1)mn)]uaub

In one imposes the linearized aether field equation (1.64) then the second and lastterms of this expression for T

(1)ab cancel, yielding

(1.67) T(1)ab = −∂0J

(1)(ab) + ∂mJ

(1)m(a ub);

J(1)ab = c1∇aub + c2ηab∇mum + c3∇bua + c4ua∇0ub

where the covariant derivatives of ua are expanded to linear order, i.e. replacedby

(1.68) (∇aub)(1) = (vb + (1/2)γ0b),a + (1/2)γab,0 − (1/2)γa0,b

Were it not for the aether background the linearized aether stress tensor (1.66)would vanish and the metric would drop out of the aether field equation, leavingall modes uncoupled.


Now diffeomorphism invariance of the action (1.60) implies that the field equa-tions are tensorial and hence covariant under diffeomorphisms. The linearizedequations inherit the linearized version of this symmetry and to find the indepen-dent physical wave modes one must fix the corresponding gauge symmetry. Thusan infinitesimal diffeomorphism generated by a vector field ξa transforms gab andua by

(1.69) δgab = Lξgab = ∇aξb +∇bξa; δua = Lξua = ξm∇mua − um∇mξa

ξa is itself first order in the perturbations so the linearized gauge transformationstake the form

(1.70) γ′ab = γab + ∂aξb + ∂bξa; v

′a = va − ∂0ξa

The usual choice of gauge in vacuum GR is the Lorentz gauge ∂aγab = 0 whereγab = γab− (1/2)γηab but for various reasons this is inappropriate here (cf. [510]).Instead one imposes directly the four gauge conditions

(1.71) γ0i = 0; vi,i = 0

To see that this is accessible note that the gauge variations of γ0i and vi,i are,according to (1.70)

(1.72) δγ0i = ξi,0 + ξ0,i; δvi,i = −ξi,i0

Thus to achieve (1.71) one must choose ξ0 and ξi to satisfy equations of the form

(1.73) (A) ξi,0 + ξ0,i = Xi (B) ξi,i0 = Y

Subtracting the second equation from the divergence of the first gives ξ0,ii =Xi,i − Y which determines ξ0 up to constants of integration by solving a Poissonequation. Then ξi can be determined up to a time independent field by integrating(A) in (1.73) with respect to time. From these choices of ξ0 and ξi (A) in (1.73)holds and the divergence of this gives (B) in (1.73). In the gauge (1.71) the tensorsin the aether and spatial metric equations in (1.64) take the forms

(1.74) J aai, = c14(vi,00 − (1/2)γ00,i0)− c1vi,kk − (1/2)c13γik,k0 − (1/2)c2γkk,0i;

G(1)ij = −(1/2)γij − (1/2)γ,ij − γk(i,j)k − (1/2)δij(γ − γ00,00 − γk,k);

T(1)ij = −c13(v(i,j)0 + (1/2)γij,00 − (1/2)c2δijγkk,00

where e.g. c14 = c1 + c4 etc. Various wave modes are calculated and in particularthere are a total of 5 modes, 2 with an unexcited aether which correspond to theusual GR modes, two “transverse” aether-metric modes, and a fifth trace aether-metric mode. We refer to [510] for details and discussion.

REMARK 5.2.1. The Lorentz violation theme and the idea of a preferredreference frame is presently of a certain general interest in connection with pro-posals of quantum gravity and we cite as before [28, 51, 76, 337, 338, 380,518, 642]. We rephrase matters here as in [380]. Thus in an effective field the-ory description the Lorentz symmetry breaking can be realized by a vector fieldthat defines the preferred frame. In the flat spacetime of the standard model thisfield can be treated as non-dynamical background structure but in the context

2. STOCHASTIC ELECTRODYNAMICS 221

of GR diffeomorphism invariance (a symmetry distinct from local Lorentz invari-ance) can be preserved by elevating this field to a dynamical quantity. This leadsto investigation of vector-tensor theories of gravity and one such model couplesgravity to a vector field that is constrained to be everywhere timelike and of unitnorm. The unit norm condition embodies the notion that the theory assigns nophysical importance to the norm of the vector and this corresponds to the Ein-stein aether (AE) theory as in [337]. In [380] one demonstrates the effect of afield redefinition on the conventional second order AE theory action. Thus takegab → g′ab = A(gab − (1 − B)uaub) with ua → (u′)a = (1/

√AB)ua where gab is a

Lorentzian metric and ua is the aether field. The action is taken as the most gen-eral form which is generally covariant, second order in derivatives, and is consistentwith the unit norm constraint. The redefinition preserves this form and the neteffect is a rescaling of the action and a transformation of the coupling constants(generalizing the work of [76]). Thus start with S = −(1/16πG)

∫ √|g|L where

(1.75) L = R + c1(∇aub)(∇aub) + c2(∇aua)(∇bub)+

+c3(∇aub)(∇bua) + c4(ua∇auc)(ub∇buc)

where R is the scalar curvature of gab with signature (+,−,−,−) and the ci aredimensionless constants. After the substitution indicated above one has

(1.76) (g′)ab =1A

(gab −

(1− 1

B

)uaub

); u′

a =√

ABua

(note (u′)au′a = 1 is preserved). The net effect of A is a rescaling of the action

by a factor of A (with the Lagrangian scaling as 1/A while the volume scales asA2). There are many calculations (omitted here) and the constructions simplifythe problem of characterizing solutions for a specific set of ci by transforming thatset into one in which one or more of the ci vanish. If non-aether matter is includeda metric redefinition not only changes the ci but also modifies the matter actionsuggesting perhaps a universal metric to which the matter couples (cf. [213, 518]).

2. STOCHASTIC ELECTRODYNAMICS

From topics in Chapters 1,2,3, and 4 we are familiar with some stochasticaspects of QM. Further the ZPF has been seen to be related to quantum phenom-ena. The idea of stochastic electrodynamcs (SED) is essentially an attempt toestablish SED as the foundation for QM. There has been some partial success inthis direction but the methods break down when trying to deal with nonlinearities.The paper [754] is a recent version of nonperturbative linear SED (LSED) whichprovides a speculative mechanism leading to the quantum behavior of field andmatter based on 3 fundamental principles; it purports to explain for example whyall systems described by it (and hence by QM ?) behave as if they consisted ofa set of harmonic oscillators. We review here first some basic background issuesin SED arising from [753] (cf. also [250, 509, 650]) and then will sketch a fewmatters from [754].


Thus start with the homogeneous Maxwell equations

(3.1) ∇ ·D = 0; −1c

∂D∂t

+∇×H = 0; ∇ ·B = 0;1c

∂B∂t

+∇×E = 0

One obtains then (we refer to [753] for details and discussion)

(3.2) B = ∇×A; E = −∇Φ− 1c

∂A∂t

; ∇×(

1c

∂A∂t

+ E)

= 0

Then the third and fourth Maxwell equations are satisfied identically and the firsttwo in combination with D = εE and B = µH determine the evolution of thepotentials A and Φ. In particular one has

(3.3) ∇2Φ +1c∂t∇ ·A = 0; ∇2A

¯− 1

c2

∂2A∂t2

−∇(∇ ·A +

1c

∂Φ∂t

)= 0

There is then room for gauge transformations

(3.4) A→ A′ = A +∇Λ; Φ → Φ′ = Φ− 1c

∂Λ∂t

Now choose the potentials to uncouple (3.3) via

(3.5) ∇ ·A +1c

∂Φ∂t

; ∇2Φ− 1c2

∂2Φ∂t2

= 0; ∇2A− 1c2

∂2A∂t2

= 0

This set is equivalent in all respects to the Maxwell equations in vacuum (Lorentzgauge) and this arrangement can always be achieved via

(3.6) ∇2Λ− 1c2

∂2Λ∂t2

= 0

Another gauge selection is the Coulomb gauge defined via

(3.7) ∇ ·A = 0; ∇2Φ = 0; ∇2A− 1c2

∂2A∂t2

=1c∇∂tΦ

One can take Φ = 0 and the last equation reduces to the last equation of (3.5). Inany case for any gauge one has

(3.8) ∇2E− 1c2

∂2E∂t2

= 0; 2A = 2B = 2E = 0

where 2 = (1/c2)∂2t − ∇2. One now goes into mode expansions with Fourier

series and integrals and we simply list formulas of the form (aα = aα(t) =anλexp(−iωnt) = anλ(t))

(3.9) qα = i

√Eα

2ω2α

(aα − a∗α); pα =

√Eα

2(aα + a∗

α);

H =∑

Hα =∑

Eαa∗αaα =

∑ 12(p2

α + ω2αq2

α)

(3.10) A =∑√

4πc2

ω2nV

eλn[pnλCos(k · x) + ωnqnλSin(k · x)];

E =∑√

4π

Veλ

n[−pnλSin(k · x) + ωnqnλCos(k · x)];


B =∑√

4πc2

ω2nV

(k × eλn)[−pnλSin(k · x) + ωnqnλCos(k · x)]

(we have written k for k ∼ kn, x for x, eλn for eλ

n, n for n ∼ (n1, n2, n3), etc. andone is thinking of a reference volume V = L1L2L3 with sides Li and ωn = c|kn|.Further kn has components ki = (2π/Li) and, setting kn = kn/|kn|, one has

(3.11) kn · eλn = 0; eλ

n · eλ′n = δλλ′ ; kn = e1

n × e2n; e1

n = −kn × e2n; e2

n = kn × e1n

Now the nonrelativistic Hamiltonian describing a charged particle in interac-tion with the radiation field is

(3.12) H =1

2m

(p− e

cA)2

+ eΦ +18π

∫d3x(E⊥2 + B2)

Here E = E⊥ + E|| and the contribution from the longitudinal part has beenwritten as the Coulomb potential eΦ; in the Coulomb gauge E|| = −∇Φ so(3.13)∫

d3E||2 =∫

d3x(∇Φ)2 =∫

d3x∇ · (Φ∇Φ)−∫

d3xΦ∇2Φ = 4π

∫d3xρ(x)Φ

since ∇2Φ = −∇·E|| = −4πρ(x) (Gauss law). From this one obtains the equationsof motion for the particle (x ∼ x)

(3.14) mx = p− e

cA; p =

e

c[x×B + (x · ∇)A]− e∇Φ

and also the following Maxwell equations in the presence of the charge and itscurrent(3.15)

∇ ·D = 4πρ; ∇×H =4π

cJ +

1c

∂D∂t

; ρ = eδ3(x− xp(t)); J = expδ3(x− xp(t))

where xp(t) is the actual position of the particle. Combining these equations oneobtains the Newton second law with the Lorentz force

(3.16) mx = eE +e

cx×B

Some simplification arises if one expresses matters via field modes (qα(t), pα(t))and (x(t), p(t)) in the form (p ∼ p, x ∼ x, etc. with e′ = e

√4π/V )

(3.17) mx = p− e

cA; p = F + e′

∑(x · eα)k

(qαcos(k · x)− pα

ωαSin(k · x)

);

qα = pα − e′(x · eα)1

ωαCos(k · x); pα = −ω2

αqα + e′(x · eα)Sin(k · x)

where F = −∇(eΦ) (without the Coulomb self-interaction). Write now

(3.18) cα(t) =1√2Eα

(pα − iωαqα); c∗α =1√2Eα

(pα + iωαqα)

Some calculation gives (x′ = x(t′))

(3.19) cα(t) = aα(t)− ie

√2π

EαVe−iωαt

∫ t

0

(x′ · eα)e−ik·x′+iωαt′dt′


Combining (3.17) and (3.19) one gets the equation of motion mx = F + F freeLor +

Fself where

(3.20) F freeLor = ie

∑√2πEα

V

(eα +

x

c× (k × eα

)× (aαe−ik·x − a∗

αeik·x);

Fself = −4πe2

V

∑(eα +

x

c× (k × eα

)×∫

dt′(x′ · eα)Cos(ωα(t′− t)− k(x′−x))

The self force is too complicated to handle here and some approximations are made(cf. [753] for details) leading to an expression

(3.21) Fself = −4e2

3c2

(12...x (t)− x(t)

π

∫ ∞

0

dω

)= mτ

...x (t)− δmx(t)

where τ = (2e2/3mc3) and δm = (4e2/3πc2)∫∞0

dω. The self radiation has twoeffects now, within the present approximation (cf. also [650]). First there is areaction force on the particle proportional to the time derivative of the accelera-tion (radiation reaction) and secondly there is a contribution to the term mx(t)involving a total or dressed mass of the particle mT = m + δm. This contributionis infinite for the point particle since

∫∞0

dω is divergent but a cure is to take acutoff ωc so that δm = (2/π)mτωc. Even for huge ωc the quantity δm/m is smallerthan 1 since τ is very small and one conjectures that in a more precise calculationthe mass correction would be at most of order α = e2/c = 1/137 (fine structureconstant). In any case one is led to the Abraham-Lorentz equation

(3.22) mT x ≡ (m + δm)x = F + F freeLor + mT τ

...x

Note that the self field terms have this simple form only in free space. In anyevent such an equation is beset with problems; as an example one looks at ahomogeneous time dependent force F (t) and the equation mx = F (t) + mτ

...x or

a− τa = F (t)/m with solution

(3.23) a(t) = et/τ

(a(0)− 1

mτ

∫ t

0

et′/τF (t′)dt′)

Problems with acausality arise and one comes to the conclusion that there seemsto be no (classical or relativistic) equation of motion for a radiating particle ininteraction with the radiation field that is free of conceptual difficulties.

A variation on the Abraham-Lorentz equation in the form

(3.24) mx = −mω20x + mτ

...x + eEx(x, t) + e

(x

c×B

)is the Braffort-Marshall equation; it is the analogue of the Langevin equation inBrownian motion (cf. here one refers back to the form mx = F +F free

Lor +Fself withm ∼ mT and τ = 2e2/3mc3). Upon approximating

...x by−ω2

0 x and linearizing onehas something tractable but we do not discuss this here. There is also considerablediscussion of harmonic oscillators and Fokker-Planck equations which we omit here.A Braffort-Marshall equation arises again in linear SED in the form

(3.25) mx = mτ...x + F (x) + eE(t)


(1-D suffices here and one observes on p.303 of [753] that SED and QM areincompatible theories). This is subsequently modified to

(3.26) mx = mτ...x + F (x) + e

∑Eka0

ke−iωt + c.c.

Here one has started from a standard Fourier representation of the ZPF in theform

(3.27) E =∑

Ekake−iωnt + c.c.; a0k ∼ eiφk ; ak → a0

k ∼ c→∞

with random phases φk uniformly distributed over (0, 2π) (x here describes theresponse of the particle to the effective field). A lot of partial averaging has goneinto this and we refer to [753] for details and discussion. Even for simple examplesthe calculations are a kind of horror story!

Let us try to summarize now some of [754]. One recalls that the centralpremise of SED is that the quantum behavior of the particle is a result of its inter-action with the vacuum radiation field or ZPF. This field is assumed to pervadethe space and is considered here to be in a stationary state with well defined sto-chastic properties. Its action on the particle is to impress upon it at every pointa stochastic motion with an intensity characterized by Planck’s constant which isa measure of the magnitude of fluctuations of the vacuum field. One begins withthe Braffort-Marshall equation in the form

(3.28) mx = f(x) + mτ...x + eE(t)

where τ = 2e2/3mc3 ∼ 10−23 for the electron. The term eE(t) stands for theelectric force exerted by the ZPF on the particle (the magnetic term is omittedsince one deals here with the nonrelativistic case). LSED is now to be based onthree principles:

(1) Principle 1. The system under study reaches an equilibrium stateat which the rate of energy radiated by the particle equals theaverage rate of energy absorbed by it from the field. To makethis quantitative multiply (3.28) by x to obtain

(3.29)⟨

dH

dt

⟩= −mτ 〈x〉+ e 〈x ·E〉

where H is the particle Hamiltonian including the Schott energy

(3.30) H =12mx2 + V (x)−mτ x · x

and V is the potential associated to the external force f . The average isover the realizations of the background ZPF and when the system hasreached the state of energetic equilibrium we have

(3.31)⟨

dH

dt

⟩= 0 ⇒ mτ

⟨x2⟩

= e 〈x ·E〉

When this equilibrium is reached (or nearly so) one says the system hasreached the quantum regime. In LSED it is claimed that detailed energybalance will hold (i.e. for every frequency). One sees that at equilibrium

the term < x > is determined by the ZPF and hence the accelerationitself should be determined by the field.

(2) Principle 2. Once the quantum regime has been attained thevacuum field has gained control over the motion of the materialpart of the system. To apply this principle consider the free particlewith mx = mτ

...x + eE(t) and express the field via

(3.32) E(t) =∑

Eβaβeiωβt =∑

ωβ>0

(E+β aβeiωβt + E−

β a∗βe−iωβt)

The aβ = aβ(ωβ) are stochastic variables and the present approach leavesthem momentarily unspecified. The amplitudes Eβ will be selected toassign to each mode of the field the mean energy Eβ = (1/2)ωβ and thisis the unique door through which Planck’s constant enters the theory.Write now

(3.33) Eβ =12

< p2β + ω2

βq2β >; pβ =

√Eβ

2(aω + a∗

ω); iωβqβ =

√Eβ

2(aω − a∗

ω)

The solution to mx = mτ...[ x] + eE(t) is then

(3.34) x(t) =∑

xβaβeiωβt = −∑ eEβaβ

mω2β + imτω3

β

eiωβt

Here all quantities except the aβ are “sure” numbers but upon intro-duction of an external force f(x) these parameters become in principlestochastic variables. Indeed from (3.28)

(3.35)∑(

−mω2βxβ − imτω2

β xβ +fβ

aβ

)= e

∑Eβaβeiωβt

For a generic force the Fourier coefficients fβ will be a complicated func-tion of (xβ) and (aβ). Writing now

(3.36) xβ = − eEβ

mω2β + imτω3

β + (fβ/xβaβ)

and putting this into (3.34) one gets

(3.37) x(t) = −∑ eEβaβ

mω2β + imτω3

β + (fβ/xβaβ)

The problem of determining x(t) in general seems impossible. One triesnow to simplify matters by looking for stable “orbits” and this leads to

(3) Principle 3. There exist states of matter (quantum states) thatare unspecific to (or basically independent of) the particularrealization of the ZPF. The ensuing calculations in [754] seem to besomewhat mysterious and we refer to this paper for further discussion(cf. also [440, 784, 858]).

3. PHOTONS AND EM 227

REMARK 5.3.1. We refer here to [177, 363, 364, 365, 366]) where, followingFeynman, the idea is to introduce QM via the relativistic theory of free photons.The arguments are very physical and historically based which would make a wel-come complement to the mainly mathematical features of the rest of this book

3. PHOTONS AND EM

We go here to [940, 941] and will concentrate on the third paper in [940].In [941] one sketches an heuristic approach to develop QM as a field theory withquantum paricles (or rather clouds) an emergent phenomenon. This is continuedin [940] for EM fields and the photon but one comes to the conclusion that itis wrong to think of the photon as the particle like duality partner of the waveassociated to the EM field. This leads to the idea that it is photons (not EM fields)that are the basic ontology and EM fields are an emergent collective property of anensemble of photons. Quantization of the fields is thus not necessary. We sketchthis now following the second and third papers in [940]. First one notes thatquantization of the Lagrangian field theory for EM presents several difficultiesand may not be the way to go (cf. [940, 982]). Second in order to maintainrelativistic covariance every Lorentz transformation must be accompanied by agauge transformation (cf. [1007]). This seems to indicate a clear preference forthinking of the photons as the entities responsible for the EM fields. The elementsof physical reality for the classical relativistic photon can be encoded in a photontensor fµν described in terms of three vectors e and b. To arrive at this oneuses gµν = gµν = diag(1,−1,−1,−1) and the completely antisymmetric tensorsεjk, εkµσρ. One postulates the existence of a massless physical system, calledphotons, transporting energy E, momentum P, and intrinsic angular momentumor spin S. The relativistic kinemmatics of a massless particle requires that its speedmust be c and the energy, momentum, and spin are related via

(4.1) E = c|P|; S×P = 0

The second relation, called the transversality constraint, follows from the fact thatthe intrinsic angular momentum of an extended object moving at the speed c cannot have any component perpendicular to the direction of propagation (easy ar-gument). Further one thinks of particles as having spatial extension since massiveparticles can not be localized beyond their Compton wavelength and a photonmoving in a well defined direction must be extended in a direction perpendicularto the direction of movement via the uncertainty principle. The massless characterof the photon suggests that its energy must be proportional to some frequency;i.e. something in the photon must be changing periodically in time. Thus formotion with pµ = (E,E, 0, 0) for example (and c = 1) a Lorentz transformationwith speed β and Lorentz factor γ = (1 − β2)−1/2 in the direction of the x1 axisleads to an energy decrease

(4.2) E′ = Eγ(1− β) = E

√1− β

1 + β

and there is a lovely interplay of physical ideas (cf. also [1021]).


in the primed reference frame. However this is precisely the transformation prop-erty of a frequency (Doppler shift) and one can therefore expect that the energyof a photon depends on some frequency (note that the argument is classical andone knows from QM that E = ν). An argument is then made in [940] that the

(4.3) fµν =

⎛⎜⎜⎝0 e1 e2 e3

−e1 0 b3 −b2

−e2 −b3 0 b1

−e3 b2 −b1 0

⎞⎟⎟⎠Note e and b are not the space components of a 4-vector but they transform as3-vectors under the subgroup of Lorentz transformations corresponding to spacerotations and translations.The energy, momentum, and spin are related by (4.1),and in any reference frame one visualizes a positive or negative helicity photon ofenergy E and spin , propagating with speed c in a direction k, as a unit vector erotating clockwise of counterclockwise in a plane orthogonal to k with frequencyω = E/. In the same plane there is another unit vector b = k× e and with thevectors e = ωe and b = ωb one can build the photon tensor fµν whose Lorentztransformations provide the description of the photon in other reference frames.The rotating vectors es, corresponding to a photon of helicity s = ±1, can begiven more conveniently via circular polarization vectors εs defined by

(4.4) ε+ =1√2(e + ib); ε− =

1√2(ie + b)

resulting in

(4.5) e+(t) = ω(eCos(ωt) + bSin(ωt)) =(

ω√2ε+e−iωt + c.c

);

e− = ω(bCos(ωt) + eSin(ωt)) =(

ω√2ε−e−iωt + c.c.

)(c.c. means complex conjugation of the previous terms). Another initial position ofthe vector can be achieved via multiplication of the circular polarization complexvectors by a phase exp(−iθ)εs to get

(4.6) es(t) =(

ω√2εse

−iωt + c.c

)The other vector bs needed to build the photon tensor is simply bs(t) = k× es(t)(note one should write εs(k) but this will be omitted). The usual orthogonalityequations are

(4.7) ε∗s · k = 0; ε∗s · εs′ = δs,s′ ; ε∗s × εs′ = iskδs,s′ ; k× εs = sε∗−s

A few other relations are

(4.8)ε− = iε∗+; εs · εs′ = i(1− δs,s′) = iδs,−s′ ; εs × εs′ = sk(1− δs,s′) = skδs,−s′

(4.9)∑s=±

(ε∗s)i(εs)j = δij − (k)i(k)j

natural object representing the photon will be a tensor (cf. [1021])

3. PHOTONS AND EM 229

where the components refer to an arbitrary set of orthogonal unit vectors (x1, x2, x3).

The QM description of a photon in a Hilbert space H = HS ⊗ HK (spin andkinematic part) is done in terms of eigenstates of fixed helicity s = ±1 and momen-tum p (in the direction k), denoted by φs,p = χk

s ⊗ φp. One takes spin operators

(4.10)

Sx =

⎛⎝ 0 0 00 0 −i0 i 0

⎞⎠ ; Sy =

⎛⎝ 0 0 i0 0 0−i 0 0

⎞⎠ ; Sz =

⎛⎝ 0 −i 0i 0 00 0 0

⎞⎠Thus (Sj)k = −iεjk and the helicity states are

(4.11)

χ± = 1

2√

1−kxky−kykz−kzkx

⎛⎝ 1− kx(kx + ky + kz)± i(ky − kz)1− ky(kx + ky + kz)± i(kz − kx)1− kz(kx + ky + kz)± i(kx − ky)

⎞⎠For the kinematic part use (rigged) square integrable functions (cf. [202]) of theform

(4.12) φp(r) =1

(√

2π)3exp

(i

p · r

)One will show now how the EM fields are built and emerge as an observableproperty of an ensemble of photons.

For many photons one builds up a Fock space with Hilbert spaces for 1, 2, · · ·times)

(4.13) a†s(p)φs1p1,··· ,snpn

=√

n + 1φsp,s1p1,··· ,snpn;

as(p)φs1p1,··· ,snpn=

1√n

n∑1

δs,siδ(p− pi)φs1p1,··· ,sipi,··· ,snpn

where sipi means that these indices are eliminated. The vacuum state φ0 withzero photons is such that as(p)φ0 = 0 and an n-photon state can be built up via

(4.14) φs1p1,··· ,snpn=

1√n!

a†s1

(p1) · · · a†sn

(pn)φ0

The symmetry requirements for identical boson states impose the commutationrelations

(4.15) [as(p), as′(p′)] = δs,s′δ3(p− p′); [a†s(p), as′(p′)] = [as(p), as′(p′)] = 0

Finally the number operator is Ns(p) = a†s(p)as(p) (number of photons with helic-

ity s and momentum in d3p centered at p) and the operator for the total numberof photons in the system is N =

∑s

∫d3pNs(p).

If we accept that the photons have objective existence and that each of them

photons with annihilation and creation operators via (we write p for p at

carries momentum p, energy E = c|p| = ω and spin in the direction of prop-agation k then for a system of many noninteracting particles the total energy,momentum, and spin are(4.16)

H =∑

s

∫d3pωNs(p); P =

∑s

∫d3ppNs(p); S =

∫d3pk(N+(p)−N−(p))

The noninteraction is a reasonably good approximation since the leading interac-tion photon-photon contribution is of fourth order in perturbation theory. Now anonhermitian operator like a†

s(p) or as(p) is related to two hermitian operators, thenumber operator (corresponding to a modulus squared) and another observable ofthe form

(4.17)∑

s

∫d3p(f(s, p, E, r, t)as(p)± c.c)

and one can guess its form. This leads to the two hermitian operators

(4.18) E(r, t) =1

2π

∑s

∫d3p√

ω(ias(p)εse(i/)(p·r−Et) + h.c.);

B(r, t) =1

2π

∑s

∫d3p√

ω(ias(p)(k× εs)e(i/)(p·r−Et) + h.c.)

One can show that these operators are indeed the EM fields and one has

(4.19) A(r, t) =c

2π

∑s

∫d3p

1√ω

(as(p)εse(i/)(p·r−Et) + h.c.);

E(r, t) = −1c∂tkA(r, t); B(r, t) = ∇×A(r, t)

(4.20) H =18π

∫d3r(E2 + B2); P =

18πc

∫d3r(E×B−B×E;

S =1

8πc

∫d3r(E×A−A×E)

(4.21) −∇×E =1c∂tB; ∇×B =

1c∂tE; ∇ ·E = 0; ∇ ·B = 0

Further, defining

(4.22) D(ρ, τ) =−1

(2π)3

∫d3pe(i/)p·ρ Sin)ωτ)

ω=

−18π2cρ

[δ)ρ− cτ)− δ(ρ + cτ)]

one finds commutator formulas for the components of E and B in terms of D(ρ, τ).The singular character of such formulas casts some question on the meaningful-ness of the EM fields as QM observables but in any event one can also calculatecommutator relations between E, B, A and the number operator N. There are anumber of interesting calculations in [940] which we omit here but we do mentionthe divergent vacuum expectation value

(4.23) < φ0,E2(r, t)φ0 >=1

(2π)2

∫d3pω

± =s 1±means )(

4. QUANTUM GEOMETRY 231

which indicates the presence of fluctuations of E in the vacuum; this is infinitebut if averaged over a small region the value would be finite. One shows also thatthe EM field of an indefinite number of photons, all with the same helicity andmomentum, is a plane wave with circular polarization; in fact for ψ

∑n Cnφn(sp)

one has

(4.24) < ψ,E(r, t)ψ >=√

ω

2π

(i∑

n

C∗nCn+1εse

(i/)p·rEt) + c.c.

)In any event it is wrong to think that the photon is the particle like dualitypartner of the wave like EM field. Another confusion analyzed in [940] is theerroneous identification of Maxwell’s equations for the EM fields with the SE forthe photon. The standard derivations of Maxwell’s equation from the SE is shownto be incorrect.

REMARK 5.4.1. We refer to [724] for further approaches to wave-particleduality and a correspondence between linearized spacetime metrics of GR andwave equations of QM.

4. QUANTUM GEOMETRY

First we sketch the relevant symbolism for geometrical QM from [54] withoutmuch philosophy; the philosophy is eloquently phrased there and in [153, 244,550, 654] for example. Thus let H be the Hilbert space of QM and write it asa real Hilbert space with a complex structure J. The Hermitian inner product isthen < φ,ψ >= (1/2)G(φ, ψ) + (i/2)Ω(φ, ψ) (note G(φ, ψ) = 2(φ, ψ) is thenatural Fubini-Study (FS) metric - cf. [244]). Here G is a positive definite realinner product and Ω is a symplectic form (both strongly nondegenerate). Moreover< φ, Jψ >= i < φ, ψ > and G(φ, ψ) = Ω(φ, Jψ). Thus the triple (J,G,Ω) equipsH with the structure of a Kahler space. Now, from [998], on a real vector space Vwith complex structure J a Hermitian form satisfies h(JX, JY ) = h(X,Y ). ThenV becomes a complex vector space via (a + ib)X = aX + bJX. A Riemannianmetric g on a manifold M is Hermitian if g(X,Y ) = g(JX, JY ) for X,Y vectorfields on M. Let ∇X be he Levi-Civita connection for g (i.e. parallel transportpreserves inner products and the torsion is zero. A manifold M with J as aboveis called almost complex. A complex manifold is a paracompact Hausdorff spacewith complex analytic patch transformation functions. An almost complex M withKahler metric (i.e. ∇XJ = 0) is called an almost Kahler manifold and if in additionthe Nijenhuis tensor vanishes it is a Kahler manifold (cf. (4.1) below). Here thedefining equations for the Levi-Civita connection and the Nijenhuis tensor are

(4.1) Γkij =

12ghk[∂igjk + ∂jgik − ∂kgji];

N(X,Y ) = [JX, JY ]− [X,Y ]− J [X,JY ]− J [JX, Y ]Further discussion can be found in [998]. Material on the Fubini-Study metricwill be provided later. Next by use of the canonical identification of the tangentspace (at any point of H) with H itself, Ω is naturally extended to a stronglynondegenerate, closed, differential 2-form on H, denoted also by Ω. The inverse ofΩ may be used to define Poisson brackets and Hamiltonian vector fields. Now in

QM the observables may be viewed as vector fields, since linear operators associatea vector to each element of the Hilbert space. Moreover the Schrodinger equation,written here as ψ = −(1/)JHψ, motivates one to associate to each quantumobservable F the vector field YF (ψ) = −(1/)JFψ. The Schrodinger vector fieldis defined so that the time evolution of the system corresponds to the flow alongthe Schrodinger vector field and one can show that the vector field YF , beingthe generator of a one parameter family of unitary mappings on H, preservesboth the metric G and the symplectic form Ω. Hence is is locally, and indeedglobally, Hamiltonian. In fact the function which generates this Hamiltonian vectorield is simply the expectation value of F . To see this write F : H → R viaF (ψ) =< ψ, Fψ >=< F >= (1/2)G(ψ, Fψ). Then if η is any tangent vector atψ

(4.2) (dF )(η) =d

dt< ψ + tη, F (ψ + tη) > |t=0 =< ψ, F η > + < η, Fψ >=

=1G(Fψ, η) = Ω(YF , η) = (iYF

Ω)(η)

where one uses the selfadjointness of F and the definition of YF (recall the Hamil-tonian vector field Xf generated by f satisfies the equation iXf

Ω = df and thePoisson bracket is defined via f, g = Ω(Xf , Xg)). Thus the time evolution ofany quantum mechanical system may be written in terms of Hamilton’s equa-tion of classical mechanics; the Hamiltonian function is simply the expectationvalue of the Hamiltonian operator. Consequently Schrodinger’s equation is simplyHamilton’s equation in disguise. For Poisson brackets we have

(4.3) F,KΩ = Ω(XF , XK) =⟨

1i

[F , K]⟩

where the right side involves the quantum Lie bracket. Note this is not Dirac’scorrespondence principle since the Poisson bracket here is the quantum one deter-mined by the imaginary part of the Hermitian inner product.

Now look at the role played by G. It enables one to define a real inner productG(XF , XK) between any two Hamiltonian vector fields and one expects that thisinner product is related to the Jordan product. Indeed

(4.4) F,K+ =

2G(XF , XK) =

⟨12[F , K]+

⟩Since the classical phase space is generally not equipped with a Riemannian metricthe Riemann product does not have a classical analogue; however it does have aphysical interpretation. One notes that the uncertainty of the observable F at astate with unit norm is (∆F )2 =< F 2 > − < F 2 >= F, F+ − F 2. Hence theuncertainty involves the Riemann bracket in a simple manner. In fact Heisenberg’suncertainty relation has a nice form as seen via

(4.5) (∆F )2(∆K)2 ≥⟨

12i

[F , K]⟩2

+⟨

12[F⊥, K⊥]+

⟩2

where F⊥ is the nonlinear operator defined by F⊥(ψ) = F (ψ)−F (ψ). Thus F⊥(ψ)is orthogonal to ψ if ‖ψ‖ = 1. Using this one can write (4.5) in the form

(4.6) (∆F )2(∆K)2 ≥(

2F,KΩ

)2

+ (F,K+ − FK)2

The last expression in (4.6) can be interpreted as the quantum covariance of F

and K.

The discussion in [54] continues in this spirit and is eminently worth reading;however we digress here for a more “hands on” approach following [189, 244, 245,246, 247, 248]. Assume H is separable with a complete orthonormal system unand for any ψ ∈ H denote by [ψ] the ray generated by ψ while ηn = (un|ψ). Definefor k ∈ N

(4.7) Uk = [ψ] ∈ P (H); ηk = 0; φk : Uk → 2(C) :

φk([ψ]) =(

η1

ηk, · · · ,

ηk−1

ηk,ηk+1

ηk, · · ·

)where 2(C) denotes square summable functions. Evidently P (H) = ∪kUk andφk φ−1

j is biholomorphic. It is easily shown that the structure is independent ofthe choice of complete orthonormal system. The coordinates for [ψ] relative to thechart (Uk, φk) are zk

n given via zkn = (ηn/ηk) for n < k and zk

n = (ηn+1/ηk) forn ≥ k. To convert this to a real manifold one can use zk

n = (1/√

2)(xkn + iyk

n) with

(4.8)∂

∂zkn

=1√2

(∂

∂xkn

+ i∂

∂ykn

);

∂

∂zkn

=1√2

(∂

∂xkn

− i∂

∂ykn

)etc. Instead of nondegeneracy as a criterion for a symplectic form inducing abundle isomorphism between TM and T ∗M one assumes here that a symplecticform on M is a closed 2-form which induces at each point p ∈ M a toplinearisomorphism between the tangent and cotangent spaces at p. For P (H) one cando more than simply exhibit such a natural symplectic form; in fact one showsthat P (H) is a Kahler manifold (meaning that the fundamental 2-form is closed).Thus one can choose a Hermitian metric G =

∑gk

mndzkm ⊗ dzk

n with

(4.9) gkmn = (1 +

∑i

zki zk

i )−1δmn − (1 +∑1

zki zk

i )−2zkmzk

n

relative to the chart Uk, φk). The fundamental 2-form of the metric G is ω =i∑

m,n gkmndzk

m ∧ dzkn and to show that this is closed note that ω = i∂∂f where

locally f = log(1 +∑

zki zk

i ) (the local Kahler function). Note here that ∂ + ∂ = dand d2 = 0 implies ∂2 = ∂2 = 0 so dω = 0 and thus P (H) is a K manifold.

Now on P (H) the observables will be represented via a class of real smoothfunctions on P (H) (projective Hilbert space) called Kahlerian functions. Considera real smooth Banach manifold M with tangent space TM, and cotangent spaceT ∗M . We remark that the extension of standard differential geometry to the infi-nite dimensional situation of Banach manifolds etc. is essentially routine modulosome functional analysis; there are a few surprises and some interesting technicalmachinery but we omit all this here. One should also use bundle terminology at

various places but we will not be pedantic about this. One hopes here to sim-ply give a clear picture of what is happening. Thus e.g. L(T ∗

x M,TxM) denotesbounded linear operators T ∗

x M → TxM and Ln(TxM,R) denotes bounded n-linearforms on TxM . An almost complex structure is provided by a smooth section Jof L(TM) = vector bundle of bounded linear operators with fibres L(TxM) suchthat J2 = −1. Such a J is called integrable if its torsion is zero, i.e. N(X,Y ) = 0with N as in (4.1). An almost Kahler (K) manifold is a triple (M,J, g) where Mis a real smooth Hilbert manifold, J is an almost complex structure, and g is a Kmetric, i.e. a Riemannian metric such that

• g is invariant; i.e. gx(JxXx, JxYx) = gx(Xx, Yx).• The fundamental two form of the metric is closed; i.e.

(4.10) ωx(Xx, Yx) = gx(JxXx, Yx)

is closed (which means dω = 0).Note that an almost K manifold is canonically symplectic and if J is integrableone says that M is a K manifold. Now fix an almost K manifold (M,J, g). Theform ω and the K metric g induce two top-linear isomorphisms Ix and Gx betweenT ∗

x M and TxM via ωx(Ixax, Xx) =< ax, Xx > and gx(Gxaz, Xx) =< ax, Xx >.Denoting the smooth sections by I, G one checks that G = J I.

Definition 5.4.1. For f, h ∈ C∞(M,R) the Poisson and Riemann bracketsare defined via f, h =< df, Idh > and ((f, h)) =< df,Gdh >. From the aboveone can reformulate this as(4.11)

f, h = ω(Idf, Idh) = ω(Gdf,Gdh); ((f, h)) = g(Gdf,Gdh) = g(Idf, Idh)

Definition 5.4.2. For f, h ∈ C∞(M,C) the K bracket is < f, h >= ((f, h))+if, h and one defines products f ν h = (1/2)ν((f, h))+fh (ν will be determinedto be ) and f ∗ν h = (1/2)ν < f, h > +fh. One observes also that

(4.12) f ∗ν h = f ν h + (i/2)νf, h; f ν h = (1/2)(f ∗ν h + h ∗ν f);

f, h = (1/iν)(f ∗ν h− h ∗ν f)

Definition 5.4.3. For f ∈ C∞(M,R) let X = Idf ; then f is called Kahlerian(K) if LXg = 0 where LX is the Lie derivative along X (recall LXf = Xf, LXY =[X,Y ], LX(ω(Y )) = (LXω)(Y )+ω(LX(Y )), · · · ). More generally if f ∈ C∞(M,C)one says that f is K if f and f are K; the set of K functions is denoted byK(M,R) or K(M,C).

Remark 5.4.1. In the language of symplectic manifolds X = df is the Hamil-tonian vector field corresponding to f and the condition LXg = 0 means that theintegral flow of X, or the Hamiltonian flow of f , preserves the metric g. Fromthis follows also LXJ = 0 (since J is uniquely determined by ω and g via (2.26)).Therefore if f is K the Hamiltonian flow of f preserves the whole K structure.Note also that K(M,R) (resp. K(M,C)) is a Lie subalgebra of C∞(M,R) (resp.C∞(M,C)).

Now P (H) is the set of one dimensional subspaces or rays of H; for everyx ∈ H/0, [x] is the ray through x. If H is the Hilbert space of a Schrodingerquantum system then H represents the pure states of the system and P (H) canbe regarded as the state manifold (when provided with the differentiable struc-ture below). One defines the K structure as follows. On P (H) one has an atlas(Vh, bh, Ch) where h ∈ H with ‖h‖ = 1. Here (Vh, bh, Ch) is the chart withdomain Vh and local model the complex Hilbert space Ch where

(4.13) Vh = [x] ∈ P (H); (h|x) = 0; Ch = [h]⊥;

bh : Vh → Ch; [x] → bh([x]) =x

(h|x)− h

This produces a analytic manifold structure on P (H). As a real manifold one usesan atlas (Vh, R bh, RCh) where e.g. RCh is the realification of Ch (the realHilbert space with R instead of C as scalar field) and R : Ch → RCh; v → Rv isthe canonical bijection (note Rv = v). Now consider the form of the K metricrelative to a chart (Vh, R bh, RCh) where the metric g is a smooth section ofL2(TP (H),R) with local expression gh : RCh → L2(RCh,R); Rz → gh

Rz where

(4.14) ghRz(Rv,Rw) = 2ν

((v|w)

1 + ‖z‖2 −(v|z)(z|w)(1 + ‖z‖2)2

)The fundamental form ω is a section of L2(TP (H),R), i.e. one can write ωh :RCh → L2(RCh,R); Rz → ωh

Rz, given via

(4.15) ωhRz(Rv,Rw) = 2ν

((v|w)

1 + ‖z‖2 −(v|z)(z|w)(1 + ‖z‖2)2

)Then using e.g. (4.14) for the FS metric in P (H) consider a Schrodinger

Hilbert space with dynamics determined via R × P (H) → P (H) : (t, [x]) →[exp(−(i/)tH)x] where H is a (typically unbounded) self adjoint operator in H.One thinks then of Kahler isomorphisms of P (H) (i.e. smooth diffeomorphismsΦ : P (H) → P (H) with the properties Φ∗J = J and Φ∗g = g). If U is anyunitary operator on H the map [x] → [Ux] is a K isomorphism of P (H). Con-versely (cf. [246]) any K isomorphism of P (H) is induced by a unitary operator U(unique up to phase factor). Further for every self adjoint operator A in H (pos-sibly unbounded) the family of maps (Φt)t∈R given via Φt : [x] → [exp(−itA)x]is a continuous one parameter group of K isomorphisms of P (H) and vice versa(every K isomorphism of P (H) is induced by a self adjoint operator where bound-edness of A corresponds to smoothness of the Φt). Thus in the present frameworkthe dynamics of QM is described by a continuous one parameter group of K iso-morphisms, which automatically are symplectic isomorphisms (for the structuredefined by the fundamental form) and one has a Hamiltonian system. Next ide-ally one can suppose that every self adjoint operator represents an observable andthese will be shown to be in 1− 1 correspondence with the real K functions.

Definition 5.4.4. Let A be a bounded linear operator on H and denote by< A > the mean value function of A defined via < A >: P (H) → C, [x] →<A >[x]= (x|Ax)/‖x‖2. The square dispersion is defined via ∆2A : P (H) →C, [x] → ∆2

[x]A =< (A− < A >[x])2 >[x].

These maps in Definition 2.4 are smooth and if A is self adjoint < A > isreal, ∆2A is nonnegative, and one can define ∆A =

√∆2A. To obtain local ex-

pressions one writes < A >h: Ch → R and (d < A >)h : Ch → (Ch)∗ via< A >h (R) = (z + h)|A(z + h))/(1 + ‖z‖2) and(4.16)

< (d < A >)hRz|Rv >= 2

(A(z + h)1 + ‖z‖2 −

(h|A(z + h))1 + ‖z‖2 h− (A(z + h)|z + h)

(1 + ‖z‖2)2 z

∣∣∣∣Rv

)Further the local expressions Xh : RCh → RCh and Y h : RCh → RCh of thevector fields X = Id < A > and Y = Gd < A > are

(4.17) Xh(Rz) = (1/ν)R(i(h|A(z + h))(z + h)− iA(z + h));

Y h(Rz) = (1/ν)R(−(h|A(z + h))(z + h) + A(z + h))One proves then (cf. [246, 446]) that the flow of the vector field X = Id < A > iscomplete and is given via Φt([x]) = [exp(−i(t/ν)A)x]. This leads to the statementthat if f is a complex valued function on P (H) then f is Kahlerian if and only ifthere is a bounded operator A such that f =< A >. From the above it is clear thatone should take ν = for QM if we want to have < H > represent Hamiltonianflow (H ∼ a Hamiltonian operator) and this gives a geometrical interpretationof Planck’s constant. The following formulas are obtained for the Poisson andRiemann brackets

(4.18) < A >,h(Rz) =

=(z + h|(1/iν)(AB −BA)(z + h))

1 + ‖z‖2 ; ((< A >,))h(Rz) =

=1ν

(z + h|(AB + BA)(z + h))1 + ‖z‖2 − 2

ν

(z + h|A(z + h))1 + ‖z‖2

(z + h|B(z + h))1 + ‖z‖2

This leads to the results(1) < A >, =< (1/iν)[A,B] >(2) ((< A >,)) = (1/ν) < AB + BA > −(2/ν) < A >; ((<

A >< A >)) = (2/ν)∆2A(3) << A >,>= (2/ν)(< AB > − < A >)(4) < A > ν = (1/2) < AB + BA >(5) < A > ∗ν =< AB >

Remark 5.4.2. One notes that (setting ν = ) item 1 gives the relation betweenPoisson brackets and commutators in QM. Further the Riemann bracket is theoperation needed to compute the dispersion of observables. In particular puttingν = in item 2 one sees that for every observable f ∈ K(P (H),R) and everystate [x] ∈ P (H) the results of a large number of measurements of f in the state[x] are distributed with standard deviation

√(/2)((f, f))([x]) around the mean

value f([x]). This explains the role of the Riemann structure in QM, namely itis the structure needed for the probabilistic description of QM. Moreover the ν

product corresponds to the Jordan product between operators (cf. item 5) anditem 4 tells us that the ∗ν product corresponds to the operator product. Thisallows one to formulate a functional representation for the algebra L(H). Thus

put ‖f‖ν =√

sup[x](f ∗ν f)([x]). Equipped with this norm K(P (H),C) becomesa W ∗ algebra and the map of W ∗ algebras between K(P (H),C and L(H) is anisomorphism. This makes it possible to develop a general functional representationtheory for C∗ algebras generalizing the classical spectral representation for commu-tative C∗ algebras. The K manifold P (H)is replaced by a topological fibre bundlein which every fibre is a K manifold isomorphic to a projective space. In particulara nonzero vector x ∈ H is an eigenvector of A if and only if d[x] < A >= 0 orequivalently if and only if [x] is a fixed point for the vector field Id < A > (inwhich case the corresponding eigenvalue is < A >[x]).

4.1. PROBABILITY ASPECTS. We go here to [33, 54, 55, 151, 153,189, 244, 247, 257, 268, 389, 408, 409, 446, 447, 612, 621, 661, 742, 765,766, 805, 937, 1005] and refer also to Section 3.1 and Remark 3.3.2. First from[151, 1005] one defines a (Riemann) metric (statistical distance) on the space ofprobability distributions P of the form

(4.19) ds2PD =

∑(dp2

j/pj) =∑

pj(dlog(pj))2

Here one thinks of the central limit theorem and a distance between probabilitydistributions distinguished via a Gaussian exp[−(N/2)(pj−pj)2/pj ] for two nearbydistributions (involving N samples with probabilities pj , pj). This can be gener-alized to quantum mechanical pure states via (note ψ ∼ √pexp(iφ) in a genericmanner)(4.20)|ψ >=

∑√pje

iφj |j >; |ψ >= |ψ > +|dψ >=∑√

pj + dpjei(φj+dφj)|j >

Normalization requires (< ψ|dψ >) = −1/2 < dψ|dψ > and measurementsdescribed by the one dimensional projectors |j >< j| can distinguish |ψ > and |ψ >according to the metric (4.19). The maximum (for optimal disatinguishability) isgiven by the Hilbert space angle cos−1(| < ψ|ψ > |) and the corresponding lineelement (PS ∼ pure state)

(4.21)14ds2

PS = [cos−1(| < ψ|ψ > |)]2 ∼ 1− | < ψ|ψ > |2 =< dψ⊥|dψ⊥ >∼

∼ 14

∑ dp2j

pj+[∑

pjdφ2j − (

∑pjdφj)2

](called the Fubini-Study (FS) metric) is the natural metric on the manifold ofHilbert space rays. Here

(4.22) |dψ⊥ >= |dψ > −|ψ >< ψ|dψ >

is the projection of |dψ > orthogonal to |ψ >. Note that if cos−1(| < ψ|ψ > | = θ

then cos(θ) = | < ψ|ψ > | and cos2(θ) = | < ψ|ψ > |2 = 1− Sin2(θ) ∼ 1− θ2 forsmall θ. Hence θ2 ∼ 1− cos2(θ) = 1− | < ψ|ψ > |2. The term in square brackets(the variance of phase changes) is nonnegative and an appropriate choice of basismakes it zero. In [151] one then goes on to discuss distance formulas in terms

of density operators and Fisher information but we omit this here. Generally asin [1005] one observes that the angle in Hilbert space is the only Riemannianmetric on the set of rays which is invariant uder unitary transformations. Inany event ds2 =

∑(dp2

i /pi),∑

pi = 1 is referred to as the Fisher metric (cf.[661]). Note in terms of dpi = pi − pi one can write d

√p = (1/2)dp/

√p with

(d√

p)2 = (1/4)(dp2/p) and think of∑

(d√

pi) as a metric. Alternatively fromcos−1(| < ψ|ψ > | one obtains ds12 = cos−1(

∑√p1i√

p2i) as a distance in P.Note from (4.21) that ds2

12 = 4cos−1| < ψ1|ψ2 > | ∼ 4(1 − |(ψ1|ψ2)|2 ≡ 4(<dψ|dψ > − < dψ|ψ >< ψ|dψ >) begins to look like a FS metric before passing toprojective coordinates. In this direction we observe from [661] that the FS metriccan be expressed also via

(4.23) ∂∂log(|z|2) = φ =1|z|2

∑dzi ∧ dzi −

1|z|4

(∑zidzi

)∧(∑

zidzi

)so for v ∼

∑vi∂i + vi∂i and w ∼

∑wi∂i + wi∂i and |z|2 = 1 one has φ(v, w) =

(v|w)− (v|z)(z|w) (cf. (3.4)).

REMARK 5.4.3. We refer now to Section 3.1 and Remark 3.3.2 for connec-tions between quantum geometry and the quantum potential via Fisher informa-tion and probability.

CHAPTER 6

INFORMATION AND ENTROPY

Information and entropy have been discussed in Sections 1.3.2, 3.3.1, 4.7, etc.and we continue with a further elaboration (see in particular [10, 23, 72, 146,173, 174, 175, 240, 343, 388, 396, 400, 431, 446, 452, 481, 512, 634, 637,639, 694, 740, 749, 755, 765, 766, 856, 914, 916, 915, 906, 976]). Asbefore we will again encounter relations to the quantum potential which serves asa persistent theme of development. There is an enormous literature on entropyand we try to select aspects which fit in with ideas of quantum diffusion andinformation theory.

1. THE DYNAMICS OF UNCERTAINTY

We begin with some topics from [396] to which we refer for certain tutorialaspects. Given events Aj (1 ≤ j ≤ N) with probabilities µj of occurance in somegame of chance with N possible outcomes one calls log(µj) an uncertainty functionfor Aj . We write the natural logarithm as log and recall that e.g. log2(b) =log(b)/ln(2) (the information theoretic base is taken as 2 in some contexts). Thequantity (Shannon entropy)

(1.1) S(µ) = −N∑1

µj log2(µj)

stands for the measure of the mean uncertainty of the possible outcomes of thegame and at the same time quantifies the mean information which is accessiblefrom an experiment (i.e. actually playing the game). Thus if one identifies theAi as labels for discrete states of a system (1.1) can be interpreted as a mea-sure of uncertainty of the state before this state is chosen and the Shannon en-tropy is a measure of the degree of ignorance concerning which possibility (eventAj) may hold true in the set of all A

′si with a given a priori probability distri-

bution (µi). Note also that 0 ≤ S(µ) ≤ log2(N) (since certainty means oneentry with probability 1 and maximum uncertainty occurs when all events areequally probable with µj = 1/N). There is some discussion of the Boltzman lawS = kBlog(W ) = −kBlog(P ) (P = 1/W ) and its relation to Shannon entropy,coarse graining, and differential entropy defined as

(1.2) S(ρ) = −∫

ρ(x)log(ρ(x))dx

(cf. Sections 1.1.6 and 1.1.8). One recalls also the vonNeumann entropy

(1.3) S(ρ) = −kBTr(ρlog(ρ))

239

240 6. INFORMATION AND ENTROPY

where ρ is the density operator for a quantum state (ρlog(ρ) is defined via func-tional calculus for selfadjoint operators (cf. [916]). For diagonal density operatorswith eigenvalues pi this will coincide with the Shannon entropy

∑pilog(pi). We go

now directly to an extension of the discussion in Sections 1.1.6-1.1.8. It is knownfrom [862] that among all one dimensional distributions ρ(x) with a finite mean,subject to the condition that the standard deviation is fixed at σ, it is the Gauss-ian with half width σ which sets a maximum of the differential entropy. Thus forthe Gaussian with ρ(x) = (1/σ

√2π)exp[−(x− x0)2/2σ2] one has

(1.4) S(ρ) ≤ 12log(2πeσ2) ⇒ 1√

2πeexp[S(ρ)] ≤ σ

A result of this is that the major role of the differential entropy is to be a mea-sure of localization in the configuration space (note that even for relatively largemean deviations σ < 1/

√2πe .26 the differential entropy S(ρ) is negative. Con-

sider now a one parameter family of probability densities ρα(x) on R whose first(mean) and second moments (variance) are finite. Write

∫xρα(x)dx = f(α) with∫

x2ραdx < ∞. Under suitable hypotheses (implying that ∂ρα/∂α is bounded bya function G(x) which together with xG(x) is integrable on R) one obtains

(1.5)∫

(x− α)2ρα(x)dx ·∫ (

∂log(ρα)∂α

)2

ραdx ≥(

df(α)dα

)2

which results from

(1.6)df

dα=∫

[(x− α)ρ1/2α ]

[∂(log(ρα))

∂αρ1/2

α

]dx

and the Schwartz inequality. Assume now that the mean value of ρα actually isα and fix at σ2 the value of the variance < (x− α)2 >=< x2 > −α2. Then (1.5)takes the familiar form

(1.7) Fα =∫

1ρα

(∂ρα

∂α

)2

dx ≥ 1σ2

where the left side is the Fisher information for ρα. This says that the Fisherinformation is a more sensitive indicator of the wave packet localization than theentropy power in (1.4). Consider now ρα = ρ(x − α) so Fα = F is no longerdependent on α and one can transform this to the QM form (up to a factor of D2

where D = /2m which we acknowledge here via the symbol ∼)

(1.8)12F =

∫1ρ

(∂ρ

∂x

)2

dx ∼∫

ρ · u2

2dx ∼ − < Q >

where u = ∇log(ρ) is the osmotic velocity field and the average < Q >=∫

ρ ·Qdx involves the quantum potential Q = 2(∆

√ρ/√

ρ) (cf. equations (6.1.13)- (6.1.16) where Q = −(2/2m)(∆

√ρ/√

ρ), Q = −(1/m)Q, D = /2m, andu = D∇log(ρ)). Consequently − < Q >≥ (1/2σ2) for all relevant probabilitydensities with any finite mean (with variance fixed at σ2). We continue in thissection with the notation Q = 2(∆

√ρ/√

ρ) and note that D2Q is in fact the correctQ = −(1/m)Q (which occasionally arises here as well, in a diffusion context).

12

1. THE DYNAMICS OF UNCERTAINTY 241

Next one defines the Kullback entropy K(θ, θ′) for a one parameter familyof probability densities ρθ so that the distance between any two densities can bedirectly evaluated. Let pθ′ be the reference density and one writes

(1.9) K(θ, θ′) = K(ρθ|ρθ′) =∫

ρθ(x)logρθ(x)ρθ′(x)

dx

(note this is positive and sometimes one refers to Hc = −K as a conditionalentropy). If one takes θ′ = θ + ∆θ with ∆θ << 1 then under a number ofstandard assumptions

(1.10) K(θ, θ + ∆θ) 12Fθ · (∆θ)2

where Fθ denotes the Fisher information measure as in (1.7). More generally fora two parameter family θ ∼ (θ1, θ2) of densities one has

(1.11) K(θ, + ∆θ) 12

∑Fij∆θi∆θj ; Fij =

∫ρθ

∂log(ρθ)∂θi

∂log(ρθ)∂θj

dx

For Gaussian densities at fixed σ with θ = α one has then K(α, α + ∆α) (∆α)2/2σ2. Various related formulas are derived and in particular one relates theShannon entropy for a coarse grained density ρB to the differential entropy of thedensity ρ leading to a formula S(ρB) − S(ρ′B) S(ρ) − S(ρ′). One considersalso spatial Markov diffusion processes in R with a diffusion coefficient D whichdrive space-time inhomogeneous probability density densities ρ(x, t). For examplea free Brownian motion characterized by v = −u = −D∇log(ρ(x, t)) and diffusioncurrent j = v · ρ obeys the continuity equation ∂tρ = −∇j which is equivalentto the heat equation. As in Sections 1.1.6-1.1.8 and 6.1 we have the importantrelations

(1.12) Q = 2D2 ∆ρ1/2

ρ1/2=

12u2 + D∇ · u; ∂tv + (v · ∇)v = −∇Q

A straightforward generalization refers to a diffusive dynamics of a mass m in aconservative force field F = −∇V . The associated Smoluchowski diffusion with aforward drift b(x) = F/mβ is analyzed in terms of a Fokker-Planck (FP) equation∂tρ = D∆ρ − ∇(b · ρ) with initial data ρ0(x) = ρ(x, 0). For standard Brownianmotion in an external force field one has D = kBT/mβ where β ∼ friction, T istemperature and kB is the Boltzman constant. With suitable hypotheses one hasthe following compatibility equations in the form of hydrodynamical conservationlaws

(1.13) ∂tρ +∇(vρ) = 0; (∂t + v · ∇)v = ∇(Ω− Q)

where Ω(x) is the volume potential for the process, namely

(1.14) Ω =12

(F

mβ

)2

+ D∇ ·(

F

mβ

)Herev = b − u = (F/mβ) − D(∇ρ/ρ) defines the current velocity of Brownianpartices in an external force field. With a solution ρ of the FP equation one asso-ciates a differential entropy S(t) = −

∫ρlog(ρ)dx which is typically not conserved.

θ

With boundary conditions on ρ, vρ, and bρ involving vanishing at boundaries orat infinity one obtains

(1.15)dS

dt=∫ [

ρ(∇ · b) + D(∇ρ)2

ρ

]dx

One emphasizes that it is not obvious whether the differential entropy grows,decreases, or whatever. One can rewrite (1.15) in the forms

(1.16) DS = D < ∇ · b > + < u2 >= D < ∇ · v >;

DS =< v2 > − = − < v · u >

where < > denotes the mean value relative to ρ. For b = F/mβ and j = vρ thisleads to a characteristic “power release” expression

(1.17)dQ

dt=

1D

∫1

mβF · jcx =

1D



Again Q can be positive (power removal) or negative (power absorption). Inthermodynamic terms one deals here with the time rate at which the mechanicalwork per unit of mass is dissipated (removed from the reservoir) in the form ofheat in the course of the Smoluchowski diffusion process - i.e. kBT ˙Q =

∫F · jdx

where T is the temperature of the bath. For b = 0 (no external forces) one hasDS = D2

∫[(∇ρ)2/ρ]dx = D2F = −D2 < Q > and one can also write

(1.18)dS

dt=(

dS

dt

)in

− dQ

dt

from (1.15) and (1.16) (here (S)in = (1/D) < v2 >.

One goes now to mean energy and the dynamics of Fisher information andconsiders −ρ and s where v = ∇s as canonically conjugate fields; then one canuse variational calculus to derive the continuity and FP equations together withthe HJ type equations whose gradient gives the hydrodynamical conservation law

(1.19) ∂ts + (1/2)(∇s)2 − (Ω− Q) = 0

Here the mean Lagrangian is

(1.20) L = −∫

ρ

[∂ts +

12(∇s)2 −

(u2

2+ Ω

)]dx

The related Hamiltonian (mean energy of the diffusion process per unit of mass)is

(1.21) H =∫

ρ

[12(∇s)2 −

(u2

2+ Ω

)]dx =

12(< v2 > − < u2 >)− < Ω >

(note here v = ∇s satisfies v = b−u with u = D∇log(ρ) and we refer to Section 1.1for clarification). One defines a thermodynamic force Fth = v/D associated withthe Smoluchowski diffusion with a corresponding potential −∇Ψ = kBTFth =F − kBT∇log(ρ) so in the absence of external forces Fth = −∇log(ρ) = −(1/D)u.The mean value of the thermodynamic force associates with the diffusion process

an analogue of the Helmholz free energy < Ψ >=< V > −TSG where the di-mensional version SG = kBS of information entropy has been introduced (it is aconfiguration space analogue of the Gibbs entropy). Here the term < V > playsthe role of (mean) internal energy and assuming ρv vanishes at boundaries (orinfinity) one obtains the time rate of change of Helmholz free energy at a constanttemperature, namely(1.22)

d

dt< Ψ >= −kBT ˙Q− T SG ⇒

d

dt< Ψ >= −(kBT )

(dS

dt

)in

= −(mβ) < v2 >

Now one can evaluate an expectation value of (1.19) which implies an identityH = − < ∂ts >. Then using Ψ = V + kBT log(ρ) (with time independent V) onearrives at Ψ = (kBT/ρ)∇(vρ) and since vρ = 0 at integration boundaries we get< Ψ >= 0. Since v = −(1/mβ)∇Ψ define then s(x, t) = (1/mβ)Ψ(x, t) so that< ∂ts >= 0 and hence H = 0 identically. This gives an interplay between themean energy and the information entropy production rate in the form

(1.23)D

2

(dS

dt

)in

=12

< v2 >=∫

ρ

(u2

2+ Ω

)dx ≥ 0

Next recalling (1.7)-(1.8) and setting F = D2Fα one obtains

(1.24) F =< v2 > −2 < Ω >≥ 0

where (1/2)F = − < Q > holds for probability densities with finite mean andvariance. One also derives the following formulas (under suitable hypotheses)

(1.25) ∂t(ρv2) = −∇ · [(ρv3)]− 2ρv · ∇(Q− Ω);

d

dt< Ω >=< v · ∇Ω >;

d

dtF =

d

dt[< v2 > −2 < Ω >] = −2 < v · ∇Q >

Then since ∇Q = ∇P/ρ where P = D2ρ∆log(ρ) (this is the real Q) the previousequation takes the form F = −

∫ρv∇Qdx = −

∫v∇Pdx which is an analogue of

the familiar expression for the power release (dE/dt) = F · v with F = −∇V inclassical mechanics.

Next in [396] there is a discussion of differential entropy dynamics in quantumtheory. Assume one has an arbitrary continuous function V(x, t) with dimensionsof energy and consider the SE in the form i∂tψ = −D∆ψ + (V/2mD)ψ. Usingψ = ρ1/2exp(is) with v = ∇s one arrives at the standard equations ∂tρ = −∇(vρ)and ∂ts + (1/2)(∇s)2 + (Ω − Q) = 0 where Ω = V/m and Q has the same formas in (1.12) (note a sign change of the Ω − Q term in comparison with (1.19)).These two equations generate a Markovian diffusion type process the probabilitydensity of which is propagated by a FP dynamics as before with drift b = v − u(instead of v = b−u) where u = D∇log(ρ) is an osmotic velocity field. Repeatingthe variational calculations one looks at (cf. (1.21))

(1.26) H =∫

ρ

[12(∇s)2 +

(u2

2+ Ω

)]dx

Then

(1.27) H = (1/2)[< v2 > + < u2 >]+ < Ω >= − < ∂ts >

For time independent V one has H = − < ∂ts >= E = const. and the FPequation propagates a probability density |ψ|2 = ρ whose differential entropy S

may nontrivially evolve in time. Maintaining the previous derivations involving(S)in one arrives at

(1.28) (S)in =2D

[E −

(12F+ < Ω >

)]≥ 0

One recalls (1/2)F = − < Q > > 0 so E− < Ω >≥ (1/2)F > 0. Hence thelocalization measure F has a definite upper bound and the pertinent wave packetcannot be localized too sharply. Note also that F = 2(E− < Ω >)− < v2 > ingeneral evolves in time (here E is a constant and Ω = 0). Using the hydrodynamicalconservation laws one sees that the dynamics of Fisher information follows the rules

(1.29)dF

dt= 2 < v∇Q >;

12F = − d

dt

[12

< v2 > + < ω >

]However F =

∫v∇Pdx where P = D2ρ∆log(ρ) and one interprets F as the mea-

sure of power transfer - keeping intact an overall mean energy H = E . We referto [396] for much more discussion and examples. We have concentrated on topicswhere the quantum potential appears in some form.

1.1. INFORMATION DYNAMICS. We go here to [173, 174, 175] andconsider the idea of introducing some kind of dynamics in a reasoning process(Fisher information can apparently be linked to semantics - cf. [907, 970]). In[173, 174] one looks at the Fisher metric defined by

(1.30) gµν =∫

X

d4xpθ(x)(

1pθ(x)

∂pθ(x)∂θµ

)(1

pθ(x)

)(∂pθ(x)∂θν

)and constructs a Riemannian geometry via

(1.31) Γσλν =

12gνσ

(∂gµν

∂θλ+

∂gλν

∂θµ− ∂gµλ

∂θν

);

Rλµνκ =

∂Γλµν

∂θκ−

∂Γλµκ

∂θν+ Γη

µνΓλκη − Γη

µκΓλνη

Then the Ricci tensor is Rµκ = Rλµλκ and the curvature scalar is R = gµκRµκ.

The dynamics associated with this metric can then be described via functionals

(1.32) J [gµν ] = − 116π

∫ √g(θ)R(θ)d4θ

leading upon variation in gµν to equations

(1.33) Rµν(θ)− 12gµν(θ)R(θ) = 0

Contracting with gµν gives then the Einstein equations Rµν(θ) = 0 (since R = 0).J is also invariant under θ → θ + ε(θ) and variation here plus contraction leadsto a contracted Bianchi identity. Constraints can be built in by adding terms(1/2)

∫ √gTµνgµνd4θ to J [gµν ]. If one is fixed on a given probability distribution


p(x) with variable θµ attached to give pθ(x) then this could conceivably describesome gravitational metric based on quantum fluctuations for example. As exam-ples a Euclidean metric is produced in 3-space via Gaussian p(x) and complexGaussians will give a Lorentz metric in 4-space. However it seems to be very re-strictive to have a fixed p(x) as the basis; it would be nice if one could vary theprobability distribution in some more general manner and study the correspondingFisher metrics (and this seems eminently doable with a Fisher metric over a spaceof probability distributions).

1.2. INFORMATION MEASURES FOR QM. We follow here [749]and derive the SE within an information theoretic framework somewhat differentfrom the exact uncertainty principle of Hall and Reginatto (cf. Sections 1.1, 3.1,and 4.7). Begin with a SE for N particles in d + 1 dimensions of the form iψt =[−(2/2m)gij∂i∂j + V ]ψ with gij = δij/m[i] where i, j = 1, · · · , dN and [i] is thesmallest integer ≥ i/d. Use the Madelung transformation ψ =

√ρexp(iS/) (cf.

[614]) to get(1.34)

∂tS +gij

2∂iS∂jS + V − 2

8gij

(2∂i∂jρ

ρ− ∂iρ∂jρ

ρ2

)= 0; ∂tρ + gij∂i(ρ∂jS) = 0

These equations can be obtained from a variational principle, minimizing the ac-tion

(1.35) Φ =∫

ρ[∂tS +

gij

2∂iS∂jS + V

]dxNddt +

2

8IF ;

IF =∫

dxNddtgijρ(∂ilog(ρ))(∂j log(ρ))

Here IF resembles the Fisher information of [369] whose inverse sets a lower boundon the variance of the probabiliy distribution ρ via the Cramer-Rao inequaliity (seeSection 1.1). (1.35) was used to derive the SE through a procedure analogous tothe principle of maximum entropy in [807, 806] (cf. also Section 1.1). Howeverthe method of [806] does not explain a priori the form of information measure thatshould be used; i.e. why must the Fisher information be minimized rather thansomething else. The aim of [749] is to construct permissible information measuresI. Thus the relevant action is

(1.36) A =∫

ρ[∂tS +

gij

2∂iS∂jkS + V

]dxNddt + λI

with λ a Lagrange multiplier. Varying this action will lead in general to a nonlinearSE

(1.37) i∂tψ =[−2

2gij∂i∂j + V

]ψ + F (ψ, ψ†)ψ

In order to have deformations of the linear theory that permit maximal preserva-tion of the usual interpretation of the wave function one considers the followingconditions:

(1) I should be real valued and positive definite for all ρ = ψ†ψ and shouldbe independent of V.


(2) I should be of the form I =∫

dxNddtρH(ρ) where H is a function ofρ(x, t) and its spatial derivatives. This will insure the weak superpositionprinciple in the equations of motion.

(3) H should be invariant under scaling, i.e. H(λρ) = H(ρ) which allowssolutions of (1.37) to be renormalized, etc.

(4) H should be separable for the case of two independent subsystems forwhich the wave function factorizes, i.e. H(ρ1ρ2) = H(ρ1) + H(ρ2).

(5) H should be Galilean invariant.(6) The action should not contain derivatives beyond second order (Absence

of higher order derivatives or AHD condition). This will insure that themultiplier λ, and hence Planck’s constant, will be the only new parame-ter that is required in making the transition from classical to quantummechanics.

The conditions 2-6 are already satisfied by the classical part of the action so it isquite minimalist to require them also of I. The homogeneity requirement 3 cannotbe satisfied if H depends only on ρ; it must contain derivatives and the AHD androtational invariance conditions imply then that H = gij(U1∂iU2∂jU3 +V1∂i∂jV2)where the Ui, Vi are functions of ρ. One can write then

(1.38) H = gij

(∂iρ∂jρ

ρ2[U1U

′2U

′3ρ

2 + V1V′′2 ρ2] +

∂i∂jρ

ρ[V1V

′2ρ]

)where the prime denotes a derivative with respect to ρ. Scaling conditions pluspositivity and universality then lead to

(1.39) I =∫

dtdxNdρgij∂iρ∂jρ

ρ2

Consequently the unique solution of the conditions 1-6 is the Fisher informationmeasure and one arrives at the linear SE since the Lagrange multiplier must thenhave the dimension of action2 thereby introducing the Planck constant. Notecondition 4 was not used but it will be useful below. Further one notes that theAHD condition ensures that within the information theoretic approach the SEis the unique single parameter extension of the classical HJ equations. One alsoargues that a different choice of metric in the information term would in fact leadback to the original gij after a nonlinear gauge transformation; this suggests that anonlinear SE is not automatically pathological. Further argument also shows thatI should not depend on S. The main difference between this and the Hall-Reginattomethod is to replace the exact uncertainly principle by condition 3.

1.3. PHASE TRANSITIONS. Referring here to [512] the introductionof a metric onto the space of parameters in models in statistical mechanics givesan alternative perspective on their phase structure. In fact the scalar curvatureR plays a central role where for a flat geometry R = 0 (noninteracting system)while R diverges at the critical point of an interacting one. Thus models arecharacterized by certain sets of parameters and given a probability distributionp(x|θ) and a sample xi the object is to estimate the parameter θ. This can bedone by maximizing the so-called likelihood function L(θ) =

∏n1 p(xi|θ) or its


logarithm. Thus one writes(1.40)

log(L(θ)) =n∑1

log(p(xi|θ)); U(θ) =d log(L(θ))

dθ; V ar[U(θ)] = −

[−d2log(L(θ))

dθ2

]The last term V ar[U(θ))] is called the expected or Fisher information and we notethat it is the same as (1.30) (see below) and in multidimensional form is expressedvia

(1.41) Gij(θ) = −E

[∂2log(p(x|θ))

∂θi∂θj

]= −

∫p(x|θ)∂2log(p(x|θ))

∂θi∂θjdx

In generic statistical-physics models one often has two parameters β (inverse tem-perature) and h (external field); in this case the Fisher-Rao metric is given byGij = ∂i∂jf where f is the reduced free energy per site and this leads to a scalarcurvature

(1.42) R = − 12G2

∣∣∣∣∣∣∂2

βf ∂β∂hf ∂2hf

∂3βf ∂2

β∂hf ∂β∂2hf

∂2β∂hf ∂β∂2

hf ∂3hf

∣∣∣∣∣∣where G = det(Gij). In some sense R measures the complexity of the system sincefor R = 0 the system is not interacting and (in all known systems) the curvaturediverges at, and only at, a phase transition point. As an example under standardscaling assumptions one can anticipate the behavior of R near a second ordercritical point. Set t = 1− (β/βc) and consider

(1.43) f(β, h) = λ−1f(tλat , hλah) = t1/atψ(ht−ah/at); at =1νd

; ah =βδ

νd

at, ah are the scaling dimensions for the energy and spin operators and d is thespace dimension. For the scalar curvature there results

(1.44) R = − 12G2

∣∣∣∣∣∣t(1/at)−2 0 t(1/at)−2(ah/at)

t(1/at)−3 0 t(1/at)−2(ah/at)−1

0 t(1/at)−2(ah/at)−1 t(1/at)−3(ah/at)

∣∣∣∣∣∣ ;G ∼ t(2/at)+2(ah/at)−2 ⇒ R ∼ ξd ∼ |β − βc|α−2

where hyperscaling (νd = 2 − α) is assumed and ξ is the correlation length. Werefer to [512] for more details, examples, and references.

1.4. FISHER INFORMATION AND HAMILTON’S EQUATIONS.Going to [755] one shows that the mathematical form of the Fisher informationI for a Gibb’s canonical probability distribution incorporates important featuresof the intrinsic structure of classical mechanics and has a universal form in termsof forces and accelerations (i.e. one that is valid for all Hamiltonians of the formT + V ). First one has shown that the Fisher information measure provides apowerful variational principle, that of extreme information, which yields most ofthe canonical Lagrangians of theoretical physics. In addition I provides an inter-esting characterization of the “arrow of time”, alternative to the one associatedwith the Boltzman entropy (cf. [776, 777]). Following [381, 384] one consid-ers a (θ, z) “scenario” in which we deal with a system specified by a physical

parameter θ while z is a stochastic variable (z ∈ RM ) and fθ(z) is a probabil-ity density for z. One makes a measurement of z and has to infer θ, callingthe resulting estimate θ = θ(z). Estimation theory states that the best possi-ble estimator θ(z), after a large number of samples, suffers a mean-square errore2 from θ that obeys a relationship involving Fisher’s I, namely Ie2 = 1, whereI(θ) =

∫dzfθ(z)[∂log(fθ(z))/∂θ]2 (only unbiased estimators with < θ >= θ are

in competition). The result here is that Ie2 ≥ 1 (Cramer-Rao bound). A caseof great importance here concerns shift invariant distribution functions where theform does not change under θ displacements and one can write

(1.45) I =∫

dzf(z)(

∂log(f(z))∂z

)2

If one is dealing with phase space where z is a M=2N dimensional vector with co-ordinates r and p then I(z) = I(r)+I(p) (cf. [755]). Now assume that one wishesto describe a classical system of N identical particles of mass m with Hamiltonian

(1.46) H = T + V =N∑1

p2i

2m+

N∑1

V (ri)

This is a simple situation but the analysis is not limited to such systems. Assumealso that the system is in equilibrium at temperature T so that in the canonicalensemble the probability density is

(1.47) ρ(r, p) =e−βH(r,p)

Z; Z =

∫d3Nrd3Np

N !h3Ne−βH(r,p)

(here for h an elementary cell in phase space one writes dτ = d3Nrd3Np/(N !h3N ,β = 1/kT with k the Boltzman constant, and Z is the partition function). Thenfrom Hamilton’s equations ∂pH = r and ∂rH = −p there results

(1.48) −kT∂log(ρ(r, p))

∂p= r; −kT

∂log(ρ(r, p))∂r

= −p

One can now write the Fisher information measure in the form(1.49)

Iτ =∫

d3Nrd3Np

N !h3Nρ(r, p)A(r, p); A = a

(∂log(ρ(r, p))

∂p

)2

+ b

(∂log(ρ(r, p))

∂r

)2

One needs two coefficients for dimensional balance (cf. [755]). One notes that

(1.50)∂log(ρ(r, p))

∂p= −β

∂H

∂p;

∂log(ρ(r, p))∂r

= −β∂H

∂r

leading to the Fisher information in the form

(1.51) (kT )2Iτ = a

⟨(∂H

∂p

)2⟩

+ b

⟨(∂H

∂r

)2⟩⇒ Iτ = β2[a < r2 > +b < p2 >]

This gives the universal Fisher form for any Hamiltonian of the form (1.46) andwe refer to [607] for connections to kinetic theory. Many other interesting resultson Fisher can be found in [381, 382, 755].

1.5. UNCERTAINTY AND FLUCTUATIONS. We go first to [38] andrecall the idea of a phase space distribution in the form (♣) µ(p, q) =< z|ρ|z >where ρ is the density matrix and |z > denotes coherent states (cf. [191, 757]for coherent states). The chosen measure of uncertainty here is the Shannoninformation

(1.52) I = −∫

dpdq

2πµ(p, q)log(µ(p, q))

The uncertainty principle manifests itself via the inequality (♠) I ≥ 1 with equalityif and only if ρ is a coherent state (cf. [609, 987]). In [38] one wants to generalizethis to include the effects of thermal fluctuations in nonequilibrium systems andwe sketch some of the ideas at least for equilibrium systems. There are in generalthree contributions to the uncertainty:

(1) The quantum mechanical uncertainty (quantum fluctuations) which isnot dependent on the dynamics.

(2) The uncertainty due to spreading or reassembly of the wave packet. Thisis a dynamical effect and it may increase or decrease the uncertainty.

(3) The uncertainty due to the coupling to a thermal environment (diffusionand dissipation).

The time evolution It of I is studied for nonequilibrium systems and it is shownto generally settle down to monotone increase. Imin

t is a measure of the amountof quantum and thermal noise the system must suffer after a nonunitary evolutionfor time t (we do not deal with this here but refer to [38] for the nonequilibriumsituation where the system decomposes into a distinguished system S plus therest, referred to as the environment; the resulting time evolution of ρ is thennonunitary). In any event the lower bound Imin

t includes the effects of 1 and 3but avoids 2.

One recalls the Shannon information (discussed earlier)

(1.53) I(S) = −N∑1

pilog(pi); 0 ≤ I(S) ≤ log(N)

This is often referred to as entropy but here the word entropy is reserved forthe vonNeumann entropy. In a similar manner, for continuous distributions (X arandom variable with probability density p(x) and

∫p(x)dx = 1), the information

of X is defined as

(1.54) I(X) = −∫

dxp(x)log(p(x))

One emphasize that p(x) here is a density (so it may be greater that 1 and I(X)may be negative). However it retains its utility as a measure of uncertainty ande.g. for a Gaussian

(1.55) p(x) =1

[2π(∆x)2]1/2exp

(− (x− x0)2

2(∆x)2

); I(X) = log

(2πe(∆x)2

)1/2

Thus I(X) is unbounded from below and goes to −∞ as ∆x→ 0 and p(x) goes toa delta function. I(X) is also unbounded from above but if the variance is fixed

then I(X) is maximized by the Gaussian distribution (1.55). Hence one has

(1.56) I(X) ≤ log(2πe(∆x)2

)1/2

The generalization to more than one variable is straightforward, e.g.

(1.57) I(X,Y ) = −∫

dxdyp(x, y)log(p(x, y)) ⇒ I(X,Y ) ≤ I(X) + I(Y )

where e.g. I(X) =∫

dyp(x, y). It is useful to introduce QM phase space distribu-tions of the form(1.58)

µ(p, q) =< z|ρ|z >; < x|z >=< x|p, q >=(

12πσ2

q

)1/4

exp

(− (x− q)2

4σ2q

+ ipx

)Here < x|z > is a coherent state with σqσp = (1/2) and there is a normalization∫

(dpdq/2π)µ(p, q) = 1. One can also show that

(1.59) µ(p, q) = 2∫

dp′dq′exp

(− (p− p′)2

2σ2p

− (q − q′)2

2σ2q

)Wρ(p′, q′);

Wρ(p, q) =1

2π

∫dξe−(i/)pξρ(q + (1/2)ξ, q − (1/2)ξ)

(Wigner function - cf. [191, 192]). One is interested in the extent to whichµ(p, q) is peaked about some region in phase space and the Shannon informationis a natural measure of the extent to which a probability distribution is peaked.Thus one takes as a measure of uncertainty the information

(1.60) I(P,Q) = −∫

dpdq

2πµ(p, q)log(µ(p, q))

One expects there to be a lower bound for I and it should be achieved on a coherentstate and this was in fact proved (cf. [609, 987]) in the form I(P,Q) ≥ 1 withequality if and only if ρ is the density matrix of a coherent state |z′ >< z′|. Further

(1.61) log( e

∆µq∆µp

)≥ I(Q) + I(P ) ≥ I(P,Q)

The variances here have the form

(1.62) (∆µq)2 = (∆ρq)2 + σ2q ; (∆µp)2 = (∆ρp)2 + σ2

p

where ∆ρ denotes the QM variance and hence

(1.63)((∆ρq)2 + σ2

q

) ((∆ρp)2 + σ2

p

)≥ 2

Minimizing (1.63) over σq (and recalling that σqσp = (1/2)) one obtains thestandard uncertainty relation ∆x∆p ≥ (/2). Now suppose one has a genuinelymixed state so that

(1.64) ρ =∑

n

pn|n >< n|; pn < 1; µ(p, q) =∑

pn| < z|n > |2

The information of (1.64) will always satisfy I(P,Q) ≥ 1 but this is a very lowlower bound; indeed from the inequality

(1.65) −(∫

dxf(x)g(x))

log

(∫dyf(y)g(y)

)≥ −

∫dxg(x)g(x)log(x)

we have(1.66)

I ≥ −∫

dpdq

2π

∑n

| < z|n > |2pnlog(pn) = −∑

pnlog(pn) = −Tr(ρlog(ρ)) ≡ S[ρ]

Thus I is bounded from below by the vonNeumann entropy S[ρ] and this is a virtueof the chosen measure of uncertainty. One sees that I is a useful measure of bothquantum and thermal fluctuations. It has a lower bound expressing the effect ofquantum fluctuations which is connected to entropy and this in turn is a measureof thermal fluctuations.

Consider now the situation of thermal equilibrium. Let the density matrix bethermal, ρ = Z−1exp(−βH) where Z = Tr(e−βH) is the partition function andβ = 1/kT . Then

(1.67) < z|ρ|z >=1Z

∑e−βEn | < z|n > |2

where |n > are energy eigenstates with eigenvalue En. For simplicity look at aharmonic oscillator for which

(1.68) H =12

(p2

M+ Mω2q2

); | < z|n > |2 =

|z|2n

n!e−|z|2 ; En = ω(n + (1/2))

Here z = (1/2)[(q/σq) + i(p/σp)] where σqσq = (1/2) and σq = (/2Mω)1/2 (cf.[191, 559, 757] for coherent states). There results

(1.69) µ(q, p) =< z|ρ|z >=(1− e−βω

)exp

(−(1− e−βω)|z|2

)The information (1.60) is then (•) I = 1 − log(1 − e−βω) which is exactly whatone expects; as T → 0 one has β → ∞ and the uncertainty reduces to the Lieb-Wehrl result I(P,Q) ≥ 1 expressing purely quantum fluctuations. For nonzerotemperature however the uncertainty is larger tending to the value −log(βω) asT → ∞ which expresses purely thermal fluctuations. It is interesting to compare(•) with the entropy S = −Tr(ρlog(ρ)). Here the partition function is Z =[2Sinh((1/2)βω)]−1 and the entropy is then S = −β(∂β(log(Z)) + log(Z) or

(1.70) S = −log[2Sinh((1/2)βω)] + (1/2)βωCoth[(1/2)βω]

For large T one has then S −log(βω) coinciding with I but S → 0 as T → 0while I goes to a nontrivial lower bound. Hence one sees that I is a useful measureof uncertainty in both the quantum and thermal regimes. We refer also to [4]where an information theoretic uncertainty relation including the effects of thermalfluctuations at thermal equilibrium has been derived using thermofield dynamics(cf. [950]); their information theoretic measure is however different than that in[38]. On goes next to non-equilibrium systems and proves for linear systems that,for each t, I has a lower bound Imin

t over all possible initial states. It coincides


with the Lieb-Wehrl bound in the absence of an environment and is related to thevonNeumann entropy in the long time limit. We refer to [38] for details.

2. A TOUCH OF CHAOS

For quantum chaos we refer to [33, 96, 97, 185, 218, 282, 359, 435, 484,491, 517, 518, 584, 659, 747, 787, 936] and begin here with [747]. Chaosis quantitatively measured by the Lyapunov spectrum of characteristic exponentswhich represent the principal rates of orbit divergence in phase space, or alter-natively by the Kolmogorov-Sinai (KS) invariant, which quantifies the rate ofinformation production by the dynamical system. Chaos is conspicuously absentin finite quantum systems but the chaotic nature of a given classical Hamiltonianproduces certain characteristic features in the dynamical behavior of its quantizedversion; these features are referred to as quantum chaos (cf. [218, 435]). They in-clude short term instabilities and diffusive behavior versus dynamical localizationand other effects. One is concerned here with an approach to the information dy-namics of the quantum-classical transition based on the HJ formalism with the KSinvariant playing a central role. The extension to the quantum domain is accom-plished via the orbits introduced by Madelung and Bohm (cf. [129, 614]); theseare natural extensions of the classical phase space flow to QM and provide therequired bridge across the transition. One striking result is that the quantum KSinvariant for a given Madelung-Bohm (MB) orbit is equal to the mean decay rateof the probability density along the orbit. Further one shows that the quantumKS invariant averaged over the ensemble of MB orbits equals the mean growthrate of configuration space information and a general and rigorous argument isgiven for the conjecture that the standard quantum-classical correspondence (orthe classical limit) breaks down for classically chaotic Hamiltonians.

We give only a sketch of results here. Thus consider a classical system of Ndegrees of freedom described by canonical variables (qi, pi) with 1 ≤ i ≤ N anddenote the Hamiltonian as H(q,p, t) with Hamilton principal function S(q, t,p0)where p0 being the initial momenta. In matrix form Hamilton’s equations areξ = J∇ξH(ξ, t) where ξ stands for the 2N dimensional phase space vector (q,p).Here J is a real antisymmetric matrix of order 2N with a 2 ⊗ N block form(0N , IN ,−IN , 0N ) which is a listing of blocks in the order (11, 12, 21, 22). Thetangent dynamics of the system is described by the 2N × 2N nonsingular matrixTµν(t, ξ0) = ∂ξµ(t, ξ0)/∂ξ0ν (the sensitivity matrix) where ξ(t, ξ0) is the trajectorystarting from ξ0 at time t0. One can in fact write (S ≡ ST - matrix transpose)

(2.1) T = (S−1p0q,−Sp0qSp0p0 , SqqS−1

p0q, Sp0q − SqqS−1p0qSp0p0)

where (Sp0q)ij = ∂2S/∂qj∂p0i. It is shown that one can write T in an upper trian-

gular block form Γ = Ω(Θ)T where Ω(Θ) = (cos(Θ),−Sin(Θ), Sin(Θ), Cos(Θ))and −σ = Tan(Θ) = −Sqq. Here Θ is a real symmetric matrix of order N whileΩ is orthogonal and symplectic (symplectic phase matrix). The upper triangularform (Γ11,Γ12, 0N ,Γ22) of Γ satisfies Γ−1

11 = Γ22 and the upper half of the Lyapunovspectrum is obtained from the singular values of Γ11 (see [747]). In particular the

2. A TOUCH OF CHAOS 253

Kolmogorov-Sinai (KS) entropy is given via

(2.2) k = limt→∞log[det(Γ11)]/t

For illustration consider the standard form H = p2/2+V (q, t) with N-dimensionalvectors q, p. Then

(2.3) k =< Tr(σ) >p.v.; < f >= limt→∞1t

∫ t

0

f(t′)dt′

where p.v. stipulates a principal value evaluation (σ will have simple pole behaviornear singularities and the principal value contribution vanishes). Since Tr(σ) =∇2

qS along the orbit (2.3) simply states that the KS invariant equals the timeaverage of the Laplacian of the action along the orbit. Now the MB formalismassociates a phase space flow with a quantum system via

(2.4) ψ = exp[iS(x, t)/ + R(x, t)]; q(t,q0,p0) = p = ∇S[q, t]

It can be verified that the expectation value of any observable in the state ψ isgiven by its average over the ensemble of orbits thus defined (e.g. Ehrenfest’sequations arise in this manner). The correspondence thus allows us to define thequantum KS invariant for a given orbit as

(2.5) k =< ∇2S >p.v.

(the averaging process is with respect to the time along the MB orbit to which Sis restricted). Now intuitively one would expect that orbits neighboring a hypo-thetical chaotic orbit in the ensemble diverge from it on the average thus causingthe orbit density along the chaotic orbit to decrease with a mean rate related tok. This is fully realized here as one sees by considering the equation of motion forR(x, t) as inherited from the SE, namely ∂tR+∇R ·∇S = −(1/2)∇2S. The char-acteristic curves for this equation are the MB orbits so that it takes the followingform along these orbits;

(2.6)dR

dt= −1

2∇2S ⇒ k = −2

⟨dR

dt

⟩= −

⟨d log(|ψ|2)

dt

⟩p.v.

This says that the quantum KS invariant for a given orbit is the mean decay rateof the probability density along the orbit. Comparing this to a classical systemwhere k = 0 while k = 0 for the quantum version one sees that the classical limitcannot hold for chaotic Hamiltonians and since chaotic classical Hamiltonians arecertainly more common than regular ones the idea of classical limit is not a reliabletest for quantum systems. Finally let k be the MB ensemble average, which is thesame as the QM expectation value, leading to

(2.7) k = limt→∞1t

∫dq|ψ|2log(|ψ|2)

which is an information entropy measure. The discussion here is very incompletebut should motivate further investigation and we refer to [747] for more detail (cf.also [976, 977] involving chaos, fractals, and entropy.


2.1. CHAOS AND THE QUANTUM POTENTIAL. The paper [745]offers an interesting perspective on the quantum potential. Thus consider a systemof n particles with the SE

(2.8) i∂tψ =

[n∑1

(−2

2mi

)∇2

i + V

]ψ; ∇i =

(∂

∂xi,

∂

∂yi,

∂

∂zi

)(here xi = (xi, yi, zi)). Set ψ = Rexp[i(S/)] and there results as usual

(2.9) ∂tS +n∑1

(∇iS)2(2mi)−1 + Q + V = 0; ∂tR2 +

n∑1

∇i ·(

R2∇iS

mi

)= 0

where Q = −∑n

1 (2/2miR)∇2i R. Now just as the causal form of the HJ equation

contains the additional term Q so the causal form of Newton’s second law containsQ as follows

(2.10) Pi = −∇iV −∇iQ; P =n∑1

Pi; P =dP

dt= −

n∑1

∇iV −n∑1

∇iQ

The author cites a number of curious and conflicting statements in the litera-ture concerning the effect of the quantum potential on Bohmian trajectories, forclarification of which he observes that for an isolated system one has

(2.11) −∑

∇iV = 0; P = 0; −∑

∇iQ = −∑

Fi = 0

Thus the sum of all the quantum forces is zero so Fi =∑n

j =i(−Fj). Thus the netquantum force on a given particle is the result of all the other particles exertingforce on this particle via the intermediary of the quantum potential. This then ishis explanation for the guidance role of the wave function.

Next it is noted that removing Q from the HJ equation is equivalent to addingthe term

(2.12)(

2

2m

)exp(iS/)∇2R =

(2

2m

)|ψ|−1ψ∇2|ψ| = −Qψ

to the SE so that the effective Hamiltonian becomes

(2.13) Heff = −(

2

2m

)∇2 + V +

(2

2m

)|ψ|−1∇2|ψ|

Since Heff = Heff (ψ) depends on ψ the superposition principle no longer applies.When φ = ψ we have

(2.14)∫

(φ∗Heffψ−ψHeffφ∗)dτ =(

2

2m

)∫φ∗ψ[|ψ|−1∇2|ψ|−|φ|−1∇2|φ|] = 0

so Heff is not Hermitian. Hence the time development operator exp[(i/)Heff t]is not unitary and the time dependent SE is a nonunitary flow. Then sincei∂t(ψ∗ψ) = ψ∗Heffψ − ψHeffψ∗ one has

(2.15) ∂t

∫|ψ − φ|2dτ =

(2

2m

)∫i[ψ∗φ− φ∗ψ][|ψ|−1∇2|ψ| − |φ|−1|φ|]dτ = 0

2. A TOUCH OF CHAOS 255

Consider then the case where two initial conditions for the time dependent SEdiffer only infinitesimally. As time progresses the two corresponding wave func-tions can become quite different, indicating the possibity of deterministic chaos,and this is a consequence of Heff being a functional of the state upon which it isacting. If the term (2/2m)|ψ|−1∇2|ψ| is removed from (2.13) one is left with aHermitian Hamiltonian and the normalization of (ψ − φ) is time independent, sothere can be no deterministic chaos. Thus in particular Q acts as a constrainingforce preventing deterministic chaos (cf. also [623]).

There are many different aspects of quantum chaos andthe perspective of [?] just mentioned does not deal with everything covered inthe references already cited (cf. also [633, 983, 1000, 1001] for additional ref-erencers). We are not expert enough to attempt any kind of in depth coveragebut extract here briefly from a few papers. First from [1000] one notes that thedBB theory of quantum motion provides motion in deterministic orbits under theinfluence of the quantum potential. This quantum potential can be very intricatebecause it generates wave interferences and further numerical work has shown thepresence of chaos and complex behavior of quantum trajectories in various systems(cf. [746]). In [1000] one indicates that movement of the zeros of the wave func-tion (called vortices) implies chaos in the dynamics of quantum trajectories. Thesevortices result from wave function interferences and have no classical explanation.In systems without magnetic fields the bulk vorticity ∇ × v in the probabilityfluid is determined by points where the phase S is singular (which can occur whenthe wave function vanishes). Due to singlevaluedness of the wave function thecirculation Γ =

∫C

rdr = (2πn/m) around a closed contour C encircling a vortexis quantized with n an integer and the velocity must diverge as one approaches avortex. This leads to a universal mechanism producing chaotic behavior of quan-tum trajectories (cf. also [746, 1001]).

Next in [983] one speaks of the edge of quantum chaos (the border betweenchaotic and non-chaotic regions) where the Lyapunov exponent goes to zero; it isthen replaced by a generalized Lyapunov coefficient describing power-law ratherthan exponential divergence of classical trajectories. In [983] one characterizesquantum chaos by comparing the evolution of an initially chosen state under thechaotic dynamics with the same state evolved under a perturbed dynamics (cf.[761]). When the initial state is in a regular region of a mixed system (one withregular and chaotic regions) the overlap remains close to one; however when theinitial state is in a chaotic zone the overlap decay is exponential. It is shown thatat the edge of quantum chaos there is a region of polynomial overlap decay. Herethe overlap is defined as O(t) = | < ψu(t)|ψp(t) > | where ψuis the state evolvedunder the unperturbed system operator and ψp is the state evolved under the per-turbed operator.

In various papers (e.g. [185, 484, 485, 517, 518, 584]) one characterizes

REMARK 6.2.1.

quantum chaos via the quantum action. This is defined via

(2.16) S[x] =∫

dtm

2x2 − V (x)

for a given classical action

(2.17) S[x] =∫

dtm

2x2 − V (x)

so that the QM transition amplitude is

(2.18) G(xf , tf ;xi, ti) = Zexp

[i

Σ∣∣∣+ xi, ti

xf ,tf

];

Σ∣∣∣xf ,tf

xi,ti

= S[xcl]∣∣∣xf ,tf

xi,ti

=∫ tf

ti

dtm

2˙x2cl − V )xcl)

∣∣∣+ xixf

where xcl is the classical path corresponding to the action S. One requires here 2-point boundary conditions xcl(t = ti) = xi and xcl(t = tf ) = xf and Z stands fora dimensionful normalisation factor. The parameters of the quantum action (i.e.mass and potential) are independent of the boundary points but depend on thetransition time T = tf − ti. A general existence proof is lacking but such quantumactions exist in many interesting cases. Then quantum chaos is defined as follows.Given a classical system with action S the corresponding quantum system displaysquantum chaos if the corresponding quantum action S in the asymptotic regimeT →∞ generates a chaotic phase space.

3. GENERALIZED THERMOSTATISTICS

We refer to [38, 262, 382, 384, 694, 696, 755, 778, 779] for discussionof various entropies based on deformed exponential functions (generalizations ofthe Boltzman-Gibbs formalism for equilibrium statistical physics), the entropiesof Beck-Cohen, Kaniadakis, Renyi, Tsallis, etc., maximum entropy ideas, escortdensity operators, and a host of other matters in generalized theormstatistics. Wesketch here first a few ideas following the third paper in [694]. Thus a model ofthermostatistics is described by a density of states ρ(E) and a probability distri-bution p(E) and for a system in thermal equilibrium at temperature T one has

(3.1) p(E) =1

Z(T )e−E/T ; Z(T ) =

∫dEρ(E)e−E/T

(Boltzman’s constant is set equal to one here). Thermal averages are definedvia < f >=

∫dEρ(E)p(E)f(E) (this is a simplified treatment with T not made

explicit - i.e. p(E) ∼ p(E, T )). A microscopic model of thermostatistics is specifiedvia an energy functional H(γ) over phase space Γ which is the set of all possiblemicrostates. Using ρ(E)dE = dγ one can write

(3.2) < f >=∫

Γ

dγp(γ)f(γ); p(γ) =e−H(γ)/T )

Z(T ); Z(T ) =

∫Γ

dγe−H(γ)/T )

In the quantum case the integration is replaced by a trace to obtain

(3.3) < f >=1

Z(T )Tr exp(−H/T )f ; Z(T ) = Tr exp(−H/T )

3. GENERALIZED THERMOSTATISTICS 257

In relevant examples of thermostatistics the density of states ρ(E) increases asa power law ρ(E) ∼ EαN with N the number of particles and α > 0. There isan energy - entropy balance where the increase of density of states ρ(E) compen-sates for the exponential decrease of probability density p(E) with a maximum ofρ(E)p(E) reached at some macroscopic energy far above the ground state energy.One can write ρ(E)p(E) = (1/Z)exp(log(ρ(E))− E/T ) with the argument of theexponential maximal when E satisfies

(3.4)1

ρ(E)ρ′(E) =

1T

where ρ′(E) is the derivative dρ/dE. If ρ(E) ∼ EαN then E ∼ αNT follows whichis the equipartition theorem. The form of the theory here indicaters that the actualform of the probability distribution is not very essential; alternative expressions forp(E) are acceptable provided they satisfy the equipartition theorem and reproducethermodynamics. One begins here by generalizing the equipartition result (3.4)and postulates the existence of an increasing positive function φ(x) defined forx ≥ 0 such that (•) (1/T ) = −[p′(E)/φ(p(E))] holds for all E and T. Then theequation for the maximum of ρ(E)p(E) becomes

(3.5) 0 =d

dE[ρ(E)p(E)] = ρ′(E)p(E)− 1

Tρ(E)φ(p(E)) ≡ ρ′(E)

ρ(E)=

1T

φ(p(E))p(E)

The Boltzman-Gibbs case is recovered when φ(x) = x. Now (•) fixes the form ofthe probability distribution p(E); to see this introduce a function logφ(x) via

(3.6) logφ(x) =∫ x

1

1φ(y)

dy

The inverse is expφ(x) and from the identity 1 = exp′φ(logφ(x))log′φ(x) there results() φ(x) = exp′φ(logφ(x)). Hence (•) can be written as

(3.7) p′(E) = − 1T

exp′φ[logφ(p(E))] ⇒ p(E) = expφ(Gφ(T )− (E/T ))

The function Gφ(T ) is the integration constant and it must be chosen so that1 =

∫dEρ(E)p(E) is satisfied. The formula (3.7) resembles the Boltzman-Gibbs

distribution but the normalization constant appears inside the function expφ(x);for φ(x) = x one has then Gφ(T ) = −log(Z(T )).

In general it is difficult to determine Gφ(T ) but an expression for its temper-ature derivative can be obtained via escort probabilities (cf. [146, 943]). Thegeneral definition is

(3.8) P (E) =1

Z(T )φ(p(E)); Z(T ) =

∫dEρ(E)φ(p(E))

Then expectation values for P (E) are denoted by

(3.9) < f >∗=∫

dEρ(E)P (E)f(E)

Note P (E) = p(E) in the Boltzman-Gibbs case φ(x) = x. Now calculate using() and (3.8) to get

(3.10)d

dTp(E) = exp′φ(Gφ(T )− (E/T ))

(d

dTGφ(T ) +

E

T 2

)=

= Z(T )P (E)(

d

dTGφ(T ) +

E

T 2

)from which follows (recall

∫dEρ(E)p(E) = 1)

(3.11) 0 =∫

dEρ(E)d

dTp(E) = Z(T )

d

dTGφ(T ) +

1T 2

Z(T ) < E >∗⇒

⇒ d

dTGφ(T ) = − 1

T 2< E >∗

Note also that combining (3.10) and (3.11) one obtains

(3.12)d

dTp(E) =

1T 2

Z(T )P (E)(E− < E >∗)

One wants now to show that generalized thermodynamics is compatible withthermodynamics begins by establishing thermal stability. Internal energy U(T ) isdefined via U(T ) =< E > with p(E) given by (3.7), so using (3.12) one obtains(3.13)

d

dTU(T ) =

∫dEρ(E)E

d

dTp(E) =

∫dEρ(E)

E

T 2Z(T )P (E)(E− < E >∗) =

=1

T 2Z(T )(< E2 >∗ − < E >2

∗) ≥ 0

Hence average energy is an increasing function of T but thermal stability requiresmore so define φ entropy (relative to ρ(E)dE via

(3.14) Sφ(p) =∫

dEρ(E)[(1− p(E))Fφ(0)− Fφ(p(E))]; Fφ(x) =∫ x

1

dylogφ(y)

One postulates that thermodynamic entropy S(T ) equals the value of the aboveentropy Sφ(p) with p given by (3.7). Then

(3.15)d

dTS(T ) =

∫dEρ(E)(−logφ(p(E))− Fφ(0))

d

dTp(E) =

=∫

dEρ(E)(−Gφ(T ) +

E

T− Fφ(0)

)d

dTp(E) =

1T

d

dTU(T )

(recall that p(E) is normalized to 1). This shows that temperature T satisfies thethermodynamic relation (1/T ) = dS/dU and since E is an increasing function ofT one concludes that S is a concave function of U; this is called thermal stability.One can also introduce the Helmholz free energy F (T ) via the well known F (T ) =U(T )− TS(T ) so from (3.15) it follows that

(3.16)d

dββF (T ) = U(T ) (β = 1/T )

Going back to (3.11) which is similar to (3.16) with F (T ) replaced by TGφ(T )and with U(T ) =< E > replaced by < E >∗ the comparison shows that TGφ(T )


is the free energy associated with the escort probability distribution P (E) up to aconstant independent of T.

The most obvious generalization now involves φ(x) = xq with q > 0 and thisessentially produces the Tsallis entropy where one has

(3.17) logq(x) =∫ x

1

dyy−q =1

1− q(x1−q − 1); expq(x) = [1 + (1− q)x]1/(1−q)

+

(cf. [?]). The probability distribution (3.17) becomes(3.18)

p(E) = [1 + (1− q)(Gq(T )− (E/T ))]1/(1−q)+ =

1zq(T )

[1− (1− q)β∗q (T )E]1/(1−q)

+

zq(T ) = (1 + (1− q)Gq(T ))1/(1−q); β∗q (T ) = zq(T )1−q/T

A nice feature of the Tsallis theory is that the correspondence between p(E) andthe escort P (E) leads to a dual structure q ↔ 1/q; indeed

(3.19) P (E) =1

Zq(T )p(E)q ⇒ p(E) =

1Z1/q(T )

P (E)1/q

Moreover there is also a q − 2 ↔ q duality; given logφ(x) a new deformed logψ(x)is obtained via

(3.20) logψ(x) = (x−1)Fφ(0)−xFφ(1/x);1

ψ(x)= Fφ(0)−Fφ(1/x)+

1x

logφ(1/x)

and for φ = xq one has ψ = (2− q)x2−q. One notes also that the definition (3.14)of entropy Sφ(p) can be written as

(3.21) Sφ(p) =∫

dEρ(E)p(E)logψ(1/p(E))

and with ψ(x) = xq

(3.22) Sq(p) =∫

dEρ(E)1

1− q(p(E)q − p(E))

3.1. NONEXTENSIVE STATISTICAL THERMODYNAMICS. Wego here to [944] for an lovely introduction and extract liberally. The Boltzman-Gibbs entropy is given via

(3.23) SBG = −k

W∑1

pilog(pi);W∑1

pi = 1

Here pi is the probability for the system to be in the ith microstate and k is theBoltzman constant kB (taken now to be 1). If every microstate has the sameprobability pi = 1/W then SBG = klog(W ). The entropy (3.23) can be shownto be nonnegative, concave, extensive, and stable (or experimentally robust). Byextensive one means that if A and B are two independent systems (i.e. pA+B

ij =pA

i pBj ) then

(3.24) SBG(A + B) = SBG(A) + SBG(B)

we get the Tsallis entropy(cf. also [1025, 1027])

One can still not derive this form of entropy (3.23) from first principles. There isalso good reason to conclude that physical entropies different from (3.23) wouldbe more appropriate for anomalous systems. In this spirit the Tsallis entropywas proposed in [945] and the property thereby generalized is extensivity. Onediscusses motivations etc. in [841] and in particular observes that the function

(3.25) y =x1−q − 1

1− q= logq(x)

satisfies

(3.26) logq(xAxB) = logq(xA) + logq(XB) + (1− q)(logq(XA))(logq(XB))

Now rewrite (3.23) in the form (k = 1)

(3.27) SBG = −W∑1

pilog(pi) =W∑1

pilog(1/pi) =⟨

log1pi

⟩The quantity log(1/pi) is called surprise or unexpectedness and one thinks of aq-surprise logq(1/pi) in defining

(3.28) Sq =⟨

logq1pi

⟩=

W∑1

pilogq(1/pi) =1−

∑W1 pq

i

q − 1

In the limit q → 1 one gets S1 = SBG and assuming equiprobability pi = 1/Wone gets

(3.29) Sq =W 1−q − 1

1− q= logq(W )

Consequently Sq is a genuine generalization of the BG entropy and the pseudo-additivity of the q-logarithm implies (restoring momentarily k)

(3.30)Sq(A + B)

k=

Sq(A)k

+Sq(B)

k+ (1− q)

Sq(A)k

Sq(B)k

if A and B are two independent systems (i.e. pA+Bij = pA

i pBj ). Thus q = 1, q < 1,

and q > 1 respectively correspond to the extensive, superextensive, and subex-tensive cases and the q-generalization of statistical mechanics is referred to asnonextensive statistical mechanics. (3.30) is true for independent A and B butif A and B are correlated in some way one can ask if extensivity would hold forsome q. For example a system whose elements are correlated at at scales mightcorrespond to W (N) ∼ Nρ ρ > 0 with entropy

(3.31) Sq(N) = logqW (N) ∼ Nρ(1−q) − 11− q

and extensivity is obtained if and only if q = 1 − (1/ρ) < 1 or Sq(N) ∝ N .Shannon and Khinchin gave early similar sets of axioms for the form of the entropyfunctional, both leading to (3.23). These were generalized in [5, 781, 780, 843]leading to the entropy

(3.32) S(p1, · · · , pW ) = k1−

∑W1 pq

i

q − 1

and it was shown that Sq is the only possible entropy extending the Boltzman-Gibbs entropy maintaining all the basic properties except extensivity for q = 1.

Some other properties are also discussed, e.g. bias, concavity, and stability.First note

(3.33) SBG = −[

d

dx

W∑1

pxi

]x=1

(x here is referred to as a bias). Similarly

(3.34) Sq = −[Dq

W∑1

pxi

]x=1

; Dqh(x) =h(qx)− h(x)

qx− 1

(Jackson derivative) and this may open the door to quantum groups (see e.g.[192]). As for concavity consider for p′′i = µpi + (1 − µ)p′i (0 < µ < 1) concavitydefined via

(3.35) S(p′′i ) ≥ µS(pi) + (1− µ)S(p′i)It can be shown that Sq is concave for every pi and q > 0. This implies the-ormdynamic stability in the framework of statistical mechanics (i.e. stability ofthe system with regard to energetic perturbations). This means that the entropyfunctional is defined such that the stationary state (thermodynamic equilibrium)makes it extreme.

There are also other generalizations of the BG entropy and we mention theRenyi entropy

(3.36) SRq =

log∑W

1 pqi

1− q=

log[1 + (1− q)Sq]1− q

and an entropy due to Landsberg, Vedral, Rajagopal, Abe defined via

(3.37) SNq = SLV RA

q =1− 1∑W

1 pqi

1− q=

Sq

1 + (1− q)Sq

These are however not concave nor experimentally robust and seem unsuited forthermodynamical purposes; on the other hand Renyi entropy seems useful forgeometrically characterizing multifractals.

Various connections of Sq to thermodynamics are indicated in [944] and wemention here first the Legendre structure. Thus for all values of q

(3.38)1T

=∂Sq

∂Uq; T =

1kβ

; Uq = − ∂

∂βlogqZq;

logqZq =Z1−q

q − 11− q

=Z1−q − 1

1− q− βUq; Fq = Uq − TSq = − 1

βlogqZq

Here Uq ∼ internal energy and Fq ∼ free energy and the specific heat is

(3.39) Cq = T∂Sq

∂T=

∂Uq

∂T= −T

∂2Fq

∂T 2

Finally a list of other properties follows supporting the thesis that Sq is a correctroad for generalizing the BG theory (see [944] for details and references); wemention a few here via

(1) Boltzmann H-theorem (macroscopic time irreversibility) q(dSq/dt) ≥0 (∀q)

(2) Ehrenfest theorem: For an observable O and a Hamiltonian H one hasd < O >q /dt = (i/) < [H, O >q (∀q)

(3) Pesin theorem (connection between sensitivity to initial conditions andthe entropy production per unit time). Define the q-generalized Kolmogorov-Sinai entropy as

(3.40) Kq = limt→∞limW→∞limN→∞< Sq > (t)

t

where N is the number of initial conditions, W is the number of windowsin the partition (fine graining), and t is discrete time (cf. also [593]).The q-generalized Lyapunov coefficient λq can be defined via sensitivityto initial conditions

(3.41) ξ = lim∆x(0)→0∆x(t)∆x(0)

= eλqtq

(focusing on a 1-D system, basically x(t+1) = g(x(t)) with g nonlinear).It was proved in [73] that for unimodal maps Kq = λq if λq > 0 andKq = 0 otherwise. More explicitly K1 = λ1 if λ1 ≥ 0 (and K1 = 0if λ1 < 0). But if λ1 = 0 then there is a special value of q such thatKq = λq if λq ≥ 0 (and Kq = 0 if λq < 0).

We refer also to [30, 31, 785, 636] for other results and approaches to thermo-dynamics, temperature, fluctuations, etc. in generalized thermostatistics and to[595] for relativistic nonextensive thermodynamics.

4. FISHER PHYSICS

The book [381] purports (with notable success) to unify several subdisciplinesof physics via Fisher information and this theme appears also in many papers, e.g.[217, 262, 239, 382, 383, 384, 385, 660, 755, 776, 777, 778, 779, 780, 781,949]. We sketch some of this here and note in passing an interesting classical-quantum trajectory in [386] which differs from a Bohmian trajectory (cf. also[93, 269, 376, 579, 629, 812, 999]). First let us sketch some summary itemsfrom [381] and then provide some details. Thus in Chapter 12 of [381] Friedenlists (among other things) the following items:

(1) Writing p = q2 in the standard formulas one can express the Fisherinformation as I = 4

∫dx(dq/dx)2 with q a real probability amplitude for

fluctuations in measurement. Under suitable conditions (see below) theinformation I obeys an I-theorem dI/dt ≤ 0. In the same spirit by whicha positive increment in therodynamic time corresponds to an increase inBoltzman entropy there is a positive increment in Fisher time defined bya decrease in information I (the two times do not always agree). Let θ be

4. FISHER PHYSICS 263

the measured phenomenon and define the Fisher temperature Tθ via

(4.1)1Tθ

= −kθ∂I

∂θ(kθ = const.)

When θ is taken to be the sysem energy E then the Fisher temperaturehas analogous properties to the ordinary Boltzman temperature, in par-ticular there is a perfect gas law pV = kETEI where p is the pressure.The I theorem can be extended to a multiparameter, multicomponentscenario with

(4.2) I = 4∫

dx∑

n

∇qn · ∇qn

(2) Any measurement of physical parameters initiates a transformation ofFisher information J → I connecting the phenomenon with the “intrin-sic data”. The phenomenological or “bound” information is denoted byJ and the acquired information is I; J is ultimately identified by an in-variance principle that characterizes the measured phenomenon. In anyexchange of information one must δJ = δI (conservation law) and forK = I − J one arrives at a variational principle (extreme physical infor-mation or EPI) K = I − J = extremum. Since J ≥ I always the EPIzero principle involves I − κJ = 0 (0 ≤ κ ≤ 1). These equations follow(independently of the axiomatic approach taken and of the I-theorem) ifthere is e.g. a unitary transformation connecting the measurement spacewith a physically meaningful conjugate space. In this manner one arrivesat the Lagrangian approach to physics, often using the Fourier transformto connect I and J. This seems a little mystical at first but many convinc-ing examples are given involving the SE, wave equations, KG equation,Dirac equation, Maxwell equations, Einstein equations, WDW equation,etc.

There is much more summary material in [381] which we omit here. A certainamount of metaphysical thinking seems necessary and Frieden remarks that JohnWheeler (cf. [988]) anticipated a lot of this in his remarks that “All things physicalare information-theoretic in origin and this is a participatory universe....Observerparticipancy gives rise to information and information gives rise to physics.” Goingnow to [381] recall I =

∫dx[(p′)2/p] = 4

∫dx(q′)2 for p = q2 and one derives the

inequality e2I ≥ 1 as follows. Look at estimators θ satisfying

(4.3) < θ(y)− θ >=∫

dy[θ(y)− θ]p(y|θ) = 0

where p(y|θ) describes fluctuations in data values y. Hence

(4.4)∫

dy(θ − θ)∂p

∂θ−∫

dyp = 0


Use now ∂θp = p(∂log(p)/∂θ) and normalization to get∫

dy(θ−θ)(∂log(p)/∂θ)p =1 which becomes(4.5)∫

dy

[∂log(p)

∂θ

√p

][(θ − θ)

√p] = 1⇒

[∫dy

(∂log(p)

∂θ

)2

p

] [∫dy(θ − θ)2p

]≥ 1

For e2 =∫

dy(θ− θ)2p this gives immediately e2I ≥ 1. One notes that if p(y|θ) =p(y−θ) then I is simply I =

∫dx(∂log(p(x))/∂x)2p(x) where x ∼ y−θ. We recall

also the Shannon entropy as H = −∫

dxp(x)log(p(x)) and the Kullback-Leiblerentropy is defined as

(4.6) G = −∫

dxp(x)logp(x)r(x)

where r(x) is a reference probability distribution function (PDF). Consider now adiscrete form of Fisher information(4.7)

I = (∆x)−1∑

n

[p(xn+1)− p(xn)]2

p(xn)= (∆x)−1

∑n

p(xn)[p(xn + ∆x)

p(xn)− 1

]2

Here p(xn+∆x)/p(xn)is close to 1 for ∆x small and one writes [p(xn+∆x)/p(xn)]−1 = ν. Then log(1 + ν) ∼ ν − (ν2/2) or ν2 = 2[ν − log(1 + ν)]. Hence I becomes

(4.8) I = −2(∆x)−1∑

n

p(xn)logp(xn + ∆x)

p(xn)+

2(∆x)−1∑

n

p(xn + ∆x)− 2(∆x)−1∑

n

p(xn)

But each of the last two terms is (∆x)−1 by normalization so they cancel leaving

(4.9) I = − 2∆x

∑p(xn)log

p(xn + ∆x)p(xn)

→ − 2∆x

G[p(x), p(x + ∆x)]

One notes (cf. [381]) that I results as a cross information between p(x) andp(x+∆x) for many different types of information measure, e.g. Renyi and Wootersinformation and in this sense serves as a kind of “mother” information. Next theI-theorem says that dI/dt ≤ 0 and this can be seen as follows. Start with (4.9) inthe form

(4.10) I(t) = −2lim∆x→0(∆x)−2

∫dx plog

p∆x

p; p∆x

= p(x + ∆x|t); p = p(x|t)

Under certain conditions (cf. [381]) p obeys a FK equation

(4.11)∂p

∂t= − d

dx[D1(x, t)p] +

d2

dx2[D2(x, t)p]

where D1 is a drift function and D2 a diffusion function. Then it is shown (cf.[776, 811]) that two PDF such as p and p∆x

that obey the FP equation have across entropy satisfying an H-theorem

(4.12) G(t) = −∫

dx plogp

p∆x

;dG(t)

dt≥ 0

Hence I obeys an I theorem dI/dt ≤ 0. We refer to [381] for more on temperature,pressure, and gas laws.

For multivariable situations one writes I = 4∫

dx∑∇qn · ∇qn with pn = q2

n.An interesting notation here is

(4.13) ψn =1√N

(q2n−1 + iq2n) (n = 1, · · · , N/2);N/2∑1

ψ∗nψn =

1N

∑q2n = p(x)

In such situations one finds for In = 4∫

dx∇qn · ∇qn and I =∑

In (cf. [381])

(4.14) In = − 2(∆x)2

Gn[pn(x|t), pn(x + ∆x|t)]; ∂In

∂t≤ 0; I(t) → min.

Now one looks at minimization problems for I where δI[q(x|t)] = 0 and for any-thing meaningful to happen the physics has to be introduced via constraints andcovariance (we refer to [381] for a more thorough discussion of these matters).Thus one is considering K = I − J and the physics is introduced via J. One canwrite e.g. I =

∫dx

∑in(x) and J =

∫dx

∑jn(x) where in = 4∇qn · ∇qn. In

general now the functional form of J follows from a statement about invariance forthe system. Examples of invariance are (i) unitary transformations such as thatbetween the space and momentum space in QM (ii) gauge invariance as in EMor gravitational theory (iii) a continuity equation for the flow, usually involvingsources. The answer q for EPI is completely dependent on the particular J(q)for that problem and that in turn depends completely on the invariance principlethat is used. If the invariance principle is not sufficiently strong in defining thesystem then one can expect the EPI output q to be only approximately correct.One has I ≤ J generally but I = J for an optimally strong invariance principle.Note κ = I/J measures the efficiency of the EPI in transferring Fisher informationfrom the phenomenon (specfied by J) to the output (specified by I). Thus κ < 1indicates that the answer q is only approximate. When the invariance principle isthe statement of a unitary transformation between the measurement space and aconjugate coordinate space then the solution to the requirement I − κJ = 0 willsimply be the reexpression of I in the conjugate space; when this holds then onecan show that in fact I = J (i.e. κ = 1). In this situation the out put q will be“correct”, i.e. not explicitly incorrect due to ignored quantum effects for example.There are in fact nonquantum and nonunitary theories for which κ = 1 (or in factany real number) and the nature of κ is not yet fully understood.

Let us call attention also to the information demon of Frieden and Soffer (cf.[381, 384]). For real Fisher coordinates x the EPI process amounts to carryingthrough a zero sum game betweem an observer (who wants to acquire maximalinformation) and an information demon (who wants to minimize the informationtransfer) with a limited resource of intrinsic information. The demon representsnature (and always wins or breaks even of course) and K = I − J ≤ 0. Furthersince ∆I = K one has ∆I ≤ 0 while ∆t ≥ 0; hence the I-theorem follows.

We run through the EPI procedure here for the KG equation which illustratesmany points. Define x1 = ix, x2 = iy, x3 = iz, x4 = ct with r = (x, y, z) and


x = (x1, x2, x3, x4) and use the ψn notation of (4.13). From I = 4∫

dx∑∇qn ·∇qn

we get

(4.15) I = 4Nc

N/2∑1

∫ ∫drdt

[−(∇ψn)∗ · ∇ψn +

(1c2

)2

(∂tψn)∗(∂tψn)

]The invariance principle here involves a unitary Fourier transformation from x toµ in the form(4.16)

(ir, ct) → (iµ/, E/c); ψn(r, t) =1

(2π)2

∫ ∫dµdEφn(µ, E)e−i(µ·r−Et)/

One recalls

(4.17)∫ ∫

drdtψ∗mψn =

∫ ∫dµdEφ∗

mφn

Differentiating in (4.16) one has (∇ψn, ∂tψn) → (−iµφn/, iEφn/) and via∇ψn ∼−iµφn/ one gets

(4.18)∫ ∫

drdt(∇ψn)∗ · ∇ψn =12

∫ ∫dµdE|φn(µ, E)|2µ2;∫ ∫

drdt(∂tψn)∗∂tψn =12

∫ ∫dµdE|φn(µ, E)|2E2

Putting this in (4.15) gives

(4.19) I =(

4Nc

2

)N/2∑1

∫ ∫dµdE|φn(µ, E)|2

(−µ2 +

E2

c2

)= J

This is the invariance principle for the given scenario. The same value of I canbe expressed in the new space (µ, E) where it is called J and J is then the bound(physical) information. Now one has from (4.17)

(4.20) c

∫ ∫drdt|ψn|2 = c

∫ ∫dµdE|φn|2 (n = 1, · · · , N/2

Summing over n and using p =∑N/2

1 ψ∗nψ = (1/N)

∑q2n with normalization gives

(4.21) 1 =∫

dµdeP (µ, E); P (µ, E) = c

N/2∑1

|φn(µ, E)|2

so P is a PDF in the (µ, E) space. One obtains then(4.22)

I = J =4N

2

∫ ∫dµdEP (µ, E)

(−µ2 +

E2

c2

); J =

(4N

2

)⟨−µ2 +

E2

c2

⟩One must have J a universal constant here so −µ2 + (E2/c2) = const. = A2(m, c)where A is some function of the rest mass m and c (which are the only otherparameters ( must also be a constant). By dimensional analysis A = mc so E2 =c2µ2 + m2c4 which links mass, momentum, and energy. This defines coordinatesµ and E as momentum and energy values. One has then I = 4N(mc/)2 = Jand the intrinsic information I in the 4-position of a particle is proportional to the


square of its intrinsic energy mc2. Since J is a universal constant (see commentsbelow), c is fixed, and given that has been fixed, one concludes that the restmass m is a universal constant. Since I measures the capacity of the observedphenomenon to provide information about (in this case) 4-length it follows that Ishould translate into a figure for the ultimate fluctuation (resolution) length that isintrinsic to QM. Here the information is I = (4N/L2) with L = /mc the reducedCompton wavelength. If all N estimates have the same accuracy some argumentthen leads to emin = L and emin corresponds to a minimal resolution length (i.e.ability to know). Finally putting things together one gets

(4.23) J =4Nm2c3

2

∫ ∫dµdE

N/2∑1

φ∗nφn =

4Nm2c3

2

∫ ∫drdt

N/2∑1

ψ∗nψ

(4.24) K = I − J =

= 4Nc

N/2∑1

∫ ∫drdt

[−(∇ψn)∗ · ∇ψn +

(1c2

)∂tψ

∗n∂tψn −

m2c2

2ψ∗

nψn

]There is much more material in [381] to enhance and refine the above ideas.

There are certain subtle features as well. In 4-dimensions the Fourier transformis unitary and covariance is achieved in all variables (treating t separately as inqn(x|t) is not a covariant formalism). EPI treats all phenomena as being statisticalin origin and every Euler-Lagrange (EL) equation determines a kind of QM forthe particular phenomenon (think here of the qn as fields). This includes classicalelectromagnetism for example where the vector potential A is considered as a kindof probability “amplitude” for photons. In 4-D the Lorentz transformation satisfiesthe requirement that Fisher information I is invariant under a change of referenceframe and this property is transmitted to J and K. Thus invariance of accuracy(or of error estimation) under a change of reference frame leads to the Lorentztransformation and to the requirement of covariance. Historically the classicalLagrangian has often been a contrivance for getting the correct answers and amain idea in [381] is to present a systematic approach to deriving Lagrangians.The Lagrangian represents the physical information k(x) =

∑kn(x), kn(x) =

in(x)− jn(x), and∫

k(x) is the total physical information K for the system. Thesolution to the variational problem for the Lagrangian can represent then (for realcoordinates) the payoff in a mathematical information game (e.g. the KG equationis a payoff expression). We exhibit now a derivation of the SE from [381] to showrobustness of the EPI scheme. The position of a particle of mass m is measured asa value y = x + θ where x is a random excursion whose probability amplitude lawq(x) is sought. Since the time t is being ignored here one is in effect looking for astationary solution to the problem. Note the issue of covariance does not arise hereand the time dependent SE is not treated since in particular it is not covariant; itcan however be obtained from the KG equation as a nonrelativistic limit. Assumethat the particle is moving in a conservative field of scalar potential V (x) withtotal energy W conserved. One defines complex wave functions as before and can

write

(4.25) I = 4NN/2∑1

∫dx

∣∣∣∣dψn(x)dx

∣∣∣∣2A Fourier transform space is defined via ψn(x) = (1/

√2π

∫dµφn(µ)exp(−iµx/)

where µ ∼ momentum. The unitary nature of this transformation guarantees thevalidity of the EPI variational procedure. One uses the Parseval theorem to get

(4.26) I =4N

2

∫dµµ2

∑n

|φn(µ)|2 = J

This corresponds to (4.19) and is the invariance principle for the given measure-ment problem. The x-coordinate expressions analogous to (4.20) and (4.21) showthat the sum in (4.26) is actually an expectation J = (4N/2) < µ2 >. Now usethe specifically nonrelativistic approximation that the kinetic energy Ekin of theparticle is µ2/2m and then

(4.27) J =8Nm

µ2< Ekin >=

8Nm

2< [W − V (x)] >=

=8Nm

2

∫dx[W − V (x)]

∑|ψn(x)|2

where the last expression is the PDF p(x). This J is the bound informationfunctional J [q] = J(ψ) and κ = 1 here. This leads to a variational problem

(4.28) K = N

N/2∑1

∫dx

[4∣∣∣∣dψn(x)

dx

∣∣∣∣2 − 8m

2[W − V (x)]|ψn(x)|2

]= extremum

The Euler-Lagrange equations are then (ψ∗nx = ∂ψ∗

n/∂x)

(4.29)d

dx

(∂L

∂ψ∗nx

)=

∂L

∂ψ∗n

; ψ′′n(x) +

2m

2[W − V (x)]ψn(x) = 0

which is the stationary SE. Since the form of equation (4.29) is the same for eachindex value n the scenario admits N = 2 degrees of freedom qn(x) or one com-plex degree of freedom ψ(x); hence the SE defines a single complex wave function.Since this derivation works with a real coordinate x the information transfer gameis being played here and the payoff is the Schrodinger wave function.

There are generalizations of EPI to nonextensive infor-mation measures in [217, 262, 239] (cf. also [755, 756, 776, 778]).

4.1. LEGENDRE THERMODYNAMICS. We go to the last paper in[382] which provides a discussion of Fisher thermodynamics and the Legendretransformation. It is shown that the Legendre transform structure of classicalthermodynamics can be replicated without change if one replaces the entropy Sby the Fisher information I. This produces a thermodynamics capable of treat-ing equilibrium and nonequilibrium situations in a traditional manner. We recallthe Shannon information measure S = −

∑P (i)log[P (i)]; it is known that if one

chooses the Boltzmann constant as the informational unit and identifies Shannon’s

REMARK 6.4.1.

entropy with the thermodynamic entropy then the whole of statistical mechan-ics can be elegantly reformulated without any reference to the idea of ensemble.The success of thermodynamics and statistical physics depends crucially on theLegendre structure and one shows now that such relationships all hold if one re-places S by the Fisher information measure. We recall that for

∫g(x, θ)dx = 1 one

writes I =∫

dxg(x, θ)[∂θg/g]2 and for shift invariant g one has I =∫

dx[(g′)2/g].There are two approaches to using Fisher information, EPI and minimum Fisherinformation (MFI), and both lead to the same results here. We write (shifting toa probability function f)

(4.30)∫

dxf(x, θ) = 1; I[f ] =∫

dxFFisher(f); FFisher(f) = f(x)[f ′/f ]2

Assume that for M functions Ai(x) the mean values < Ai > are known where

(4.31) < Ai >=∫

dxAi(x)f(x)

This represents information at some appropriate (fixed) time t. The analysis willuse MFI (or EPI) to find the probability distribution fI = fMFI that extremizesI subject to prior conditions < Ai > and the result will be given via solutions ofa stationary Schrodinger like equation. The Fisher based extremization problemhas the form (F (f) = FFisher(f))

(4.32) δf

[I(f)− α < 1 > −

M∑1

λi < Ai >

]= 0 ≡

δf

[∫dx

(F (f)− αf −

M∑1

λiAif

)]= 0

Variation leads to ((α, λ1, · · · , λM ) are Lagrange multipliers)

(4.33)∫

dxδf

[(f)−2

(∂f

∂x

)2

+∂

∂x

(2f

∂f

∂x

)+ α +

M∑1

λiAi

]= 0

and on account of the arbitrariness of δf this yields

(4.34) (f)−2(f ′)2 +∂

∂x(2/f)f ′) + α +

M∑1

λiAi = 0

The normalization condition on f makes α a function of the λi and we assumefI(x, λ) to be a solution of (4.34) where λ ∼ (λi). The extreme Fisher informationis then

(4.35) I =∫

dxf−1I (∂xfI)2

Now to find a general solution of (4.34) define G(x) = α +∑M

1 λiAi(x) and write(4.34) in the form

(4.36)[∂log(fI)

∂x

]2

+ 2∂2log(fI)

∂x2+ G(x) = 0

Make the identification fI = (ψ)2 now the introduce a new variable v(x) =∂log(ψ(x))/∂x. Then (4.36) becomes

(4.37) v′(x) = −[G(x)

4+ v2(x)

]which is a Riccati equation. This leads to

(4.38) u(x) = exp

[∫ x

dx[v(x)]]

= exp

[∫ x

dxdlog(ψ)

dx

]= ψ;

−12ψ′′(x)− 1

8

M∑1

λiAi(x)ψ(x) =α

8ψ(x)

where the Lagrange multiplier α/8 plays the role of an energy eigenvalue and thesum of the λiAi(x) is an effective potential function U(x) = (1/8)

∑M1 λiAi(x).

We note (in keeping with the Lagrangian spirit of EPI) that the Fisher informa-tion measure corresponds to the expectation value of the kinetic energy of theSE. Note also that (4.38) has multiple solutions and it is reasonable to supposethat the solution leading to the lowest I is the equilibrium one. Now standardthermodynamics uses derivatives of the entropy S with respect to λi and < Ai >and we start from (4.35) and write after an integration by parts

(4.39)∂I

∂λi=∫

dx∂fI

∂λi

[−f−2

I (f ′I)

2 − ∂

∂x

(2fI

f ′I

)]Comparing this to (4.34) one arrives at

(4.40)∂I

∂λi=∫

dx∂fI

∂λi

[α +

M∑1

λjAj

]which on account of normalization yields

(4.41)∂I

∂λi=

M∑1

λj∂

∂λi

∫dxfIAj(x) ≡ ∂I

∂λi=

N∑1

λj∂

∂λi< Aj >

This is a generalized Fisher-Euler theorem whose thermodynamic counterpart isthe derivative of the entropy with respect to the mean values. One computes easily

(4.42)∑

i

∂I

∂λi

∂λi

∂ < Aj >=∑

i

∑k

λk∂ < Ak >

∂λi

∂λi

∂ < Aj >⇒ ∂I

∂ < Aj >= λj

as expected. The Lagrange multipliers and mean values are seen to be conjugatevariables and one can also say that fI = fI(λ1, · · · , λM ).

Now as the density fI formally depends on M + 1 Lagrange multipliers,normalization

∫dxfI(x) = 1 makes α a function of the λi and we write α =

α(λ1, · · · , λM ). One can assume that the input information refers to the λi andnot to the < Ai >. Introduce then a generalized thermodynamic potential (Le-gendre transform of I) as

(4.43) λJ(λ1, · · · , λM ) = I(< A1 >, · · · , < AM >)−M∑1

λi < Ai >

Then

(4.44)∂λJ

∂λi=

M∑1

∂I

∂ < Aj >

∂ < Aj >

∂λi−

M∑1

λj∂ < Aj >

∂λi− < Ai) = − < Ai >

where (4.42) has been used. Thus the Legendre structure can be summed up in

(4.45) λJ = I −M∑1

λi < Ai >;∂λJ

∂λi= − < Ai >;

∂I

∂ < Ai >= λi;

∂λi

∂ < Aj >=

∂λj

∂ < Ai >=

∂2I

∂ < Ai > ∂ < Aj >;

∂ < Aj >

∂λi=

∂ < Ai >

∂λj= − ∂2λJ

∂λi∂λj

As a consequence one can recast (4.41) in the form

(4.46)∂I

∂λi=

M∑1

λj∂

∂λj< Ai >

Thus the Legendre transform structure of thermodynamics is entirely translatedinto the Fisher context.

4.2. FIRST AND SECOND LAWS. We go here to [779] where one showsthe coimplication of the first and second laws of thermodynamics. Thus macro-scopically in classical phenomenological thermodynamics the first and second lawscan be regarded as independent statements. In statistical mechanics an underly-ing microscopic substratum is added that is able to explain thermodynamics itself.Of this substratum a microscopic probability distribution (PD) that controls thepopulation of microstates is a basic ingredient. Changes that affect exclusively mi-crostate population give rise to heat and how these changes are related to energychanges provides the essential content of the first law (cf. [809]). In [779] oneshows that the PD establishes a link between the first and second laws accordingto the following scheme.

• Given: An entropic form (or an information measure) S, a mean energyU and a temperature T, and for any system described by a microscopicPD pi a heat transfer process via pi → pi + dpi then

• If the PD pi maximizes S this entails dU = TdS and alternatively• If dU = TdS then this predetermines a unique PD that maximizes S.

For the second law one wants to maximize entropy S with M appropriate con-straints Ak which take values Ak(i) at the microstate i; the constrains have theform

(4.47) < Ak >=∑

i

piAk(i) (k = 1, · · ·M)

The Boltzman constant is kB and assume that k = 1 in (4.47) corresponds to theenergy E with A1(i) = εi so that the above expression specializes to

(4.48) U =< A1 >=∑

piεi

One should now maximize the “Lagrangian” Φ given by

(4.49) Φ =S

kB− α

∑i

pi − β∑

i

piεi −M∑2

λk

∑i

piAk(pi)

in order to obtain the actual distribution pi from the equation δpiΦ = 0. Since here

one is interested just in the “heat” part the last term on the right of (4.49) willnot be considered. It is argued that if pi changes to pi + dpi because of δpi

Φ = 0one will have

(4.50) 0 =dS

kB− βdU

(note∑

i δpi = 0 via normalization). Since β = 1/kBT we get dU = TdS soMaxEnt implies the first law.

The central goal here is to go the other way so assume one has a rather generalinformation measure of the form

(4.51) S = k∑

i

pif(pi)

where k ∼ kB . The sum runs over a set of quantum numbers denoted by i (char-acterizing levels of energy εi) that specify an appropriate basis in Hilbert space,P = pi is an (as yet unknown) probability distribution with

∑pi = constant,

and f is an arbitrary smooth function of the pi. Assume further that mean valuesof quantities A that take the value Ai with probability pi are evaluated via

(4.52) < A >=∑

i

Aig(pi)

In particular the mean energy U is given by U =∑

i εig(pi). Assume now thatthe set P changes in the fashion

(4.53) pi → pi + dpi;∑

dpi = 0

(the last via∑

pi = constant. This in turn generates corresponding changes dSand dU and one is thinking here of level population changes, i.e. heat. To insurethe first law one assumes (•) dU − TdS = 0 and as a consequence of (•) a littlealgebra gives (up to first order in the dpi the condition

(4.54) εig′(pi)− kT [f(pi) + pif

′(pi)] = 0

This equation is now examined for several situations. First look at Shannon en-tropy with

(4.55) f(pi) = −log(pi); g(pi) = pi

In this situation (4.54) becomes

(4.56) −εi = kT [log(pi) + 1] ⇒ pi =1eexp(−εi/kT )

After normalization this is the canonical Boltzmann distribution and this is theonly distribution that guarantees obedience to the first law for Shannon’s informa-tion measure. A posteriori this distribution maximizes entropy as well with U asa constraint which establishes a link with the second law. Several other measures


are considered, in particular the Tsallis measure, and we refer to [779] for details.In summary if one assumes entropy is maximum one immediately derives the firstlaw and if you assume the first law and an information measure this predeterminesa probability distribution that maximizes entropy.

There is currently a great interest in acoustic wave phe-nomena, sound and vortices, acoustic spacetime, acoustic black holes, etc. A primesource of material involves superfluid physics a la Volovik [968, 969] and Bose-Einstein condensates (see e.g. [39, 40, 86, 81, 101, 115, 369, 370, 912, 913,963]). We had originally written out material from [912, 913] in preparation forsketching some material from [968]. However we realized that there is simply toomuch to include in this book; at least 2-3 more chapters would be needed to evenget off the ground.

REMARK 6.4.2.

CHAPTER 7

ON THE QUANTUM POTENTIAL

1. RESUME

We have seen already how the quantum potential arises in many contexts asa fundamental ingredient connected with quantum matter, and how it provideslinkage between e.g. statistics and uncertainty, Fisher information and entropy,Weyl geometry, and quantum Kahler geometry. We expand further on certainaspects of the quantum potential in this Chapter aafter the resume from Chapters1-6 Moreover we have seen how the trajectory representation a la deBroglie-Bohm(dBB) can be used to develop meaningful insight and results in quantum fieldtheory (QFT) and cosmology. This is achieved mainly without the elaborate ma-chinery of Fock spaces, Feynman diagrams, operator algebras, etc. in a straight-forward manner. The conclusion seems inevitable that dBB theory is essentiallyall pervasive and represents perhaps the most powerful tool available for under-standing not only QM but the universe itself. There are of course many papersand opinions concerning such conclusions (some already discussed) and we willmake further comments along these lines in this Chapter. We remark that thereis some hesitation in postulating an ensemble interpretation when using dBB the-ory in cosmology with a wave function of the universe for example but we see noobstacle here, once an ensemble of universes is admitted. This is surely as reason-able as dealing with many string theories as is now fashionable. In any event we

1.1. THE SCHRODINGER EQUATION. We list some examples pri-marily concerned with QM.

(1) The SE in 1-dimension with ψ = Rexp(iS/) and associated HJ andcontinuity equations are

(1.1) − 2

2mψxx + V ψ = iψt; St +

S2x

2m+ V + Q = 0;

Q = −2R′′

2mR; ∂t(R2) +

1m

(R2Sx)x = 0

Here P = R2 = |ψ|2 is a probability density, p = mx = Sx (with Snot constant!) is the momentum, ρ = mP is a mass density, and Qis the quantum potential of Bohm. Classical mechanics involves a HJequation with Q = 0 and can be derived as follows (cf. [304]). Considera Lagrangian L = px−H with Hamiltonian H = (p2/2m)+V and action

275

begin with a resume of highlights from Chapters 1-6 and gather some material→here (see also [1026] for information on the map SE Q).

276 7. ON THE QUANTUM POTENTIAL

S =∫ t2

t1dtL(x, x, t). One computes

(1.2) δS =∫ t2

t1

dt

[p

d

dtδx + xδp− δH −H

d

dtδt

]where δH = Hxδx + Hpδp + Htδt = Vxδx + (p/m)δp + Htδt. Then

(1.3) δS =∫ t2

t1

dt

[p

d

dtδx + xδp− Vxδx− p

mδp−Htδt−H

d

dtδt

]=

=∫ t2

t1

dt

d

dt[pδx−Hδt] + δp

(x− p

m

)− δx

(dp

dt+ Vx

)+ δt[H −Ht]

Consequently, writing δS = (Sxδx + Stδt)|t2t1 one arrives at

(1.4) x =p

m; p = −Vx; H = Ht; p = Sx; St + H = 0

Note here the “surface” term (from integration) is G = pδx − Hδt andδS = G2 − G1 which should equal δS = (∂S/∂x1)δx1 + (∂S/∂x2)δx2 +(∂S/∂t1)δt1 + (∂S/∂t2)δt2 where x1 = x(t1) and x2 = x(t2) One seesthen directly how the addition of Q to a classical HJ equation producesa quantum situation.

(2) Another classical connection comes via hydrodynamics (cf. Section 1.1)where (1.1) can be put in the form

(1.5) ∂t(ρv) + ∂(ρv2) +ρ

m(∂V + ∂Q) = 0

which is like an Euler equation in fluid mechanics moduo a pressureterm −ρ−1∂P on the right. If we identify (ρ/m)∂Q = ρ−1∂P ≡ P =∂−1(ρ2/m)∂Q (with some definition of ∂−1 - cf. [205]) then the quantumterm could be thought of as providing a pressure with Q correspondinge.g. to a stress tensor of a quantum fluid. We refer also to Remark 1.1.2and work of Kaniadakis et al where the quantum state coressponds toa subquantum statistical ensemble whose time evolution is governed byclassical kinetics in phase space.

(3) The Fisher information connection a la Remarks 1.1.4 - 1.1.5 involves aclassical ensemble with particle mass m moving under a potential V

(1.6) St +1

2m(S′)2 + V = 0; Pt +

1m

∂(PS′)′ = 0

where S is a momentum potential; note that no quantum potential ispresent but this will be added on in the form of a term (1/2m)

∫dt(∆N)2

in the Lagrangian which measures the strength of fluctuations. Thiscan then be specified in terms of the probability density P as indicatedin Remark 1.1.4 leading to a SE where by Theorem 1.1.1 (∆N)2 ∼c∫

[(P ′)2/P ]dx. A “neater” approach is given in Remark 1.1.5 leading in1-D to

(1.7) St +1

2m(S′)2 + V +

λ

m

((P ′)2

P 2− 2P ′′

P

)= 0

Note that Q = −(2/2m)(R′′/R) becomes for R = P 1/2 an equationQ = −(2/8m)[(2P ′′/P ) − (P ′/P )2]. Thus the addition of the Fisher

1. RESUME 277

information serves to quantize the classical system via a quantum po-tential and this gives a direct connection of the quantum potential withfluctuations.

(4) The Nagasawa theory (based in part on Nelson’s work) is very revealingand fascinating. The essense of Theorem 1.1.2 is that ψ = exp(R + iS)satisfies the SE iψt + (1/2)ψ′′ + iaψ′ − V ψ = 0 if and only if

(1.8) V = −St +12R′′ +

12(R′)2 − 1

2(S′)2 − aS′; 0 = Rt +

12S′′ + S′R′ + aR′

Changing variables (X = (/√

m)x and T = t) one arrives at iψT =−(2/2m)ψXX − iAψX + V ψ where A = a/

√m and

(1.9) iRT + (2/m2)RXSX + (2/2m2)SXX + ARX = 0;

V = −iST + (2/2m)RXX + (2/2m2)R2X − (2/2m2)S2

X −ASX

The diffusion equations then take the form

(1.10) φT +2

2mφXX + AφX + cφ = 0; −φT +

2

2mφXX −AφX + cφ = 0;

c = −V (X,T )− 2ST −2

mS2

X − 2ASX

It is now possible to introduce a role for the quantum potential in thistheory. Thus from ψ = exp(R + iS) (with = m = 1 say) we have ψ =ρ1/2exp(iS) with ρ1/2 = exp(R) or R = (1/2)log(ρ). Hence (1/2)(ρ′/ρ) =R′ and R′′ = (1/2)[(ρ′′/ρ) − (ρ′/ρ)2] while the quantum potential isQ = (1/2)(∂2ρ1/2/ρ1/2) = −(1/8)[(2ρ′′/ρ) − (ρ′/ρ)2]. Equation (1.8)becomes then

(1.11) V = −St +18

(2ρ′′

ρ− (ρ′)2

ρ2

)− 1

2(S′)2 − aS′ ≡

≡ St +12(S′)2 +V +Q+aS′ = 0; ρt +ρS′′ +S′ρ′ +aρ′ = 0 ≡ ρt +(ρS′)′ +aρ′ = 0

Thus −2St − (S′)2 = 2V + 2Q + 2aS′ and one has

PROPOSITION 1.1. The creation-annihilation term c in the dif-fusion equations (cf. Theorem 1.1.2) becomes

(1.12) c = −V − 2St − (S′)2 − 2aS′ = V + 2Q

where Q is the quantum potential.

(5) Going to Remarks 1.1.6-1.1.8 we set

(1.13) Q =2

2m2

∂2√ρ√

ρ= − 1

mQ; D =

2m; u = D∂(log(ρ)) =

h

2m

ρ′

ρ

Then u is called an osmotic velocity field and Brownian motion involvesv = −u for the diffusion current. In particular

(1.14) Q =12u2 + D∂u


One defines an entropy term S = −∫

ρlog(ρ)dx leading, for suitableregions of integration and behavior of ρ at infinity, and using ρt = −∂(vρ)from (1.1), to

(1.15)∂S

∂t= −

∫ρt(1 + log(ρ)) =

∫(1 + log(ρ))∂(vρ) =

= −∫

vρ′ =∫

uρ′ = D

∫(ρ′)2

ρ

Note also

(1.16) Q =D2

2

(2ρ′′

ρ−(

ρ′

ρ

)2)⇒

∫ρQ = −D2

2

∫(ρ′)2

ρ

Thus generally ∂S/∂t ≥ 0 and F = −(2/D2)∫

ρQ is a functional formof Fisher information with St = DF.

(6) The development of the SE by Nottale, Cresson, et al in Section 1.2 isbasically QM and is peripheral to scale relativity as such. The idea isroughly to to imagine e.g. continuous nondifferentiable quantum pathsand to describe the velocity in terms of an average V = (1/2)(b+ + b−)and a discrepancy U = (1/2)(b+− b−) where b± are given by (2.1). Not-tale’s derivation is heuristic but revealing and working with a complexvelocity he captures the complex nature of QM. In particular the quan-tum potential can be written as Q = −(m/2)U2−(/2)∂U correspondingto (1.14) via Q = −(1/m)Q. As indicated in Proposition 1.2.1 this re-veals the quantum potential as a manifestation of the “fractal” natureof quantum paths - smooth paths correspond to Q = 0 which seems topreclude smooth trajectories for quantum particles. In such a case thestandard formula x = (/2m)[ψ∗∂ψ/|ψ|2] requires a discontinuous xwich places some constraints on ψ and the whole guidance idea. Thiswhole matter should be addressed further along with considerations ofosmotic velocity, etc. Another approach to quantum fractals is given inSection 1.5.

(7) In section 2.3.1 we sketched some of the Bertoldi-Faraggi-Matone (BFM)version of Bohmian mechanics and in particular for the stationary quan-tum HJ equation (QHJE) (1/2m)S2

x + W = E (W = V + Q), arisingfrom −(2/2m)ψ′′ + V ψ = Eψ, one can extract from [347] the formulasfor trajectories (using Floydian time t ∼ ∂S/∂E). Thus (∂EW = ∂EQ)

(1.17) t ∼ ∂E

∫Sxdx = ∂E

∫[E −W ]1/2dx =

(m

2

)1/2∫

(1− ∂EQ√E −W

dx

Hence

(1.18)dt

dx=(m

2

)1/2 1− ∂EQ√E −W

⇒ x =Sx

m

11− ∂EQ

Thus m(1 − ∂EQ)x = mQx = Sx and this is defined as p with mQ

representing a quantum mass. Note x = p/m and we refer to [194,191, 347, 373, 374] for discussion of all this. Further via p′ = m′

Qx +

1. RESUME 279

mQ(x/x), etc., one can rewrite the QSHJ as a third order trajectory (ormicrostate) equation (see also Remark 7.4)

(1.19)m2

Q

2mx2 + V − E +

2

4m

(m′′

Q

mQ− 3

2

(m′

Q

mQ

)2

−m′

Q

mQ

x

x2+

...x

x3− 5

2x2

x4

)= 0

In Remark 2.2.2 with Theorem 2.1 we observed how the uncertainty prin-ciple of QM can be envisioned as due to incomplete information aboutmicrostates when working in the Hilbert space formulation of QM basedon the SE. It was shown how ∆q∆x = O() arises automatically from aBFM perspective. Thus the canonical QM in Hilbert space cannot see asingle trajectory and hence is obliged to operate in terms of ensemblesand probability. We have also seen how a probabilistic ensemble picturewith quantum fluctuations comes about with the fluctuations correspond-ing to the quantum potential (see Item 3 above). This suggests that abackground motivation for the Hilbert space may really exist (beyondits pragmatic black magic) since these fluctuations represent a form ofinformation (and uncertainty). The hydrodynamic and diffusion modelsare also directly connected to this and produce as in Item 5 above aconnection to entropy.

1.2. DEBROGLIE-BOHM. There are many approaches to dBB theoryand in fact much of the book is concerned with this. David Bohm wrote exten-sively about the subject but we have omitted much of the philosophy (implicateorder, etc.). The book by Holland [471] is excellent and a modern theory is be-ing constructed by Durr, Goldstein, Zanghi, et al (cf. also the work of Bertoldi,Farragi, Matone, and Floyd). Some new directions in QFT, Weyl geometry, andcosmology are also covered in the book, due to Barbosa, Pinto-Neto, Nikolic, A.and F. Shojai, et al, and we will try to summarize some of that here.

(1) The BFM theory is quite novel (and profound) in that it is based en-tirely on an equivalence principle (EP) stating that all physical systemscan be connected by a coordinate transformation to the free situationwith vanishing energy. One bases the stationary situation of energy Ein the nonrelativistic case with the SE as in Item 7 above. In the rela-tivistic case (Remark 2.2.3) one can work in the same spirit directly witha Minkowski metric to obtain the Klein-Gordon (KG) equation with arelativistic quantum potential

(1.20) Qrel = − 2

2m

R

R

Note that the probability aspects concerning R appear to be absent in therelativistic theory. It is interesting to note (cf. Remark 2.2.1) that theEP implies that all mass can be generated by a coordinate transformationand since mass can be expressed in terms of the quantum potential Qthis provides yet another role for the quantum potential.

(2) Some quantum field theory (QFT) aspects of the Bohm theory are de-veloped in [471] and sketched here in Example 2.1.1. One arrives at a


formula, namely

(1.21) ψ = − δQ[ψ(x), t]δψ(x)

∣∣∣∣ψ(x)=ψ(x,t)

; Q[ψ, t] = − 12R

∫d3x

δ2R

δψ2

More recently there have been some impressive papers by Nikolic in-volving Bohmian theory and QFT. First there are papers on bosonicand fermionic Bohmian QFT sketched in Sections 3.2 and 3.3. Theseare lovely but even more attractive are two newer papers [708, 713] byNikolic which we displayed in Sections 2.4 and 2.6. In [708] one utilizesthe deDonder-Weyl formulation of QFT (reviewed in Appendix A) anda Bohmian formulation is not postulated but derived from the technicalrequirements of covariance and consistency with standard QM. One in-troduces a preferred foliation of spacetime with Rµ normal to the leaf Σand writes R([φ],Σ) =

∫Σ

dΣµRµ with S([φ], x) =∫Σ

dΣµSµ. This pro-duces a covariant version of Bohmian mechanics with Ψ = Rexp(iS/)via

(1.22)12

dSµ

dφ

dSµ

dφ+ V + Q + ∂µSµ = 0;

dRµ

dφ

dSµ

dφ+ J + ∂µRµ = 0

(1.23) Q = − 2

2R

δ2R

δΣφ2(x); J =

R

2δ2S

δΣφ2(x)

In [713] one uses the many fingered time (MF) Tomonaga-Schwinger(TS) equation where a Cauchy hypersurface Σ is defined via x0 = T (x)with x corresponding to coordinates on Σ. The TS equation is

(1.24) iδΨ[φ, T ]δT (x)

= HΨ[φ, T ]

Take a free scalar field for convenience with

(1.25) H(x) = −12

δ2

δφ2(x)+

12[(∇φ(x))2 + m2φ2(x)]

Then for a manifestly covariant theory one introduces parameters s =(s1, s2, s3) to serve as coordinates on a 3-dimensional manifold Σ in space-time with xµ = Xµ(s) the embedding coordinates. The induced metricon Σ is

(1.26) qij(s) = gµν(X(s))∂Xµ(s)

∂si

∂Xν(s)∂sj

Similarly a normal (resp. unit normal - transforming as as a spacetimevector) to the surface are

(1.27) n(s) = εµαβγ∂Xα

∂s1

∂Xβ

∂s2

∂Xγ

∂s3; nµ(s) =

gµν nν√|gαβnαnβ |

Then for x→ s and δδT (x) → nµ(s) δ

δXµ(s) the TS equation becomes

(1.28) H(s)Ψ[φ,X] = inµ(s)δΨ[φ,X]δXµ(s)

1. RESUME 281

and the Bohmian equations of motion are (Ψ = Rexp(iS) )

(1.29)∂Φ(s, T ]∂τ(s)

=1

|q(s)|1/2

δS

δφ(s)

∣∣∣∣φ=Φ

;∂

∂τ(s)≡ limσx→0

∫σx

d3snµ(s)δ

δXµ(s)

In the same spirit the quantum MFT KG equation is

(1.30)

[(∂

∂τ(s)

)2

+∇i∇i + m2

]Φ(s, X] = − 1

|q(s)|1/2

∂Q(s, φ,X]∂φs)

∣∣∣∣φ=Φ

where ∇i is the covariant derivative with respect to si and

(1.31) Q(s, φ,X] = − 1|q(s)|1/2

12R

δ2R

δφ2(s)

(3) The QFT model in Section 2.5 involving stochastic jumps is quite techni-cal and should be read in conjunction with Nagasawa’s book [674]. Theidea (cf. [326]) is that for the Hamiltonian of a QFT there is associateda |ψ|2 distributed Markov process, typically a jump process (to accoundfor creation and annihilation processes) on the configuration space of avariable number of particles. One treats this via functional analysis, op-erator thery, and probability, which leads to mountains of detail, only asmall portion of which is sketched in this book.

(4) In Section 3.2 we give a sketch of dBB in Weyl geometry following A.and F. Shojai [873]. This is a lovely approach and using Dirac-Weylmethods one is led comfortably into general relativity (GR), cosmology,and quantum gravity, in a Bohmian context. Such theories dominateChapters 3 and 4. First one looks at the relativistic energy equationηµνpµpν = m2c2 generalized to

(1.32) ηµνPµP ν = m2c2(1 +Q) = M2c2; Q = (2/m2c2)(|Ψ|/|Ψ|)

(1.33) M2 = m2

(1 + α

|Ψ||Ψ|

); α =

2

m2c2

(obtained e.g. by setting ψ =√

ρexp(iS/) in the KG equation). HereM2 is not positive definite and in fact (1.32) is the wrong equation!Some interesting arguments involving Lorentz invariance lead to betterequations and for a particle in a curved background the natural quantumHJ equation is most comfortably phrased as

(1.34) ∇µ(ρ∇µS) = 0; gµν∇µS∇νS = M2c2; M

2 = m2eQ; Q =2

m2c2

g|Ψ||Ψ|

This is equivalent to

(1.35)(

m2

M2

)gµν∇µS∇νS = m2c2

showing that the quantum effects correspond to a change in spacetimemetric gµµ → gµν = (M2/m2)gµν . This is a conformal transformationand leads to Weyl geometry where (1.35) takes the form gµν∇µS∇νS =


m2c2 with ∇µ the covariant derivative in the metric gµν . The particlemotion is then

(1.36) Md2xµ

dτ2+ MΓµ

νκuνuκ = (c2gµν − uµuν)∇νM

and the introduction of a quantum potential is equivalent to introduc-ing a conformal factor Ω2 = M2/m2 in the metric (i.e. QM corre-sponds to Weyl geometry). One considers then a general relativisticsystem containing gravity and matter (no quantum effects) and links itto quantum matter by the conformal factor Ω2 (using an approximation1 + Q ∼ exp(Q) for simplicity); then the appropriate Einstein equationsare written out. Here the conformal factor and the quantum potentialare made into dynamical fields to create a scalar-tensor theory with twoscalar fields. Examples are developed and we refer to Section 3.2 and[873] for more details. Back reaction effects of the quantum factor onthe background metric are indicated in the modified Einstein equations.Thus the conformal factor is a function of the quantum potential and themass of a relativistic particle is a field produced by quantum correctionsto the classical mass. In general frames both the spacetime metric andthe mass field have quantum properties.

(5) The Dirac-Weyl theory is developed also in [873] via the action

(1.37) A =∫

d4x√−g(FµνFµν − β2 WR+ (σ + 6)β;µβ;µ + Lmatter

The gravitational field gµν and Weyl field φµ plus β determine the space-time geometry and one finds a Bohmian theory with β ∼ M (Bohmianquantum mass field). We will say much more about Dirac-Weyl theorybelow.

(6) There is an interesting approach by Santamato in [840, 841] dealing withthe SE and KG equation in Weyl geometry (cf. Section 3.3 and [189,203]). In the first paper on the SE one assumes particle motion givenby a random process qi(t, ω) with probability density ρ(q, t), qi(t, ω) =vi(q(t, ω), t), and random initial conditions qi

0(ω) (i = 1, · · · , n). One be-gins with a stochastic construction of (averaged) classical type Lagrangeequations in generalized coordinates for a differentiable manifold M inwhich a notion of scalar curvature R is meaningful (this is where statis-tics enters the geometry). It is then shown that a theory equivalent to QM(via a SE) can be constructed where the “quantum force” (arising froma quantum potential Q) can be related to (or described by) geometricproperties of space. To do this one assumes that a (quantum) Lagrangiancan be constructed in the form L(q, q, t) = LC(q, q, t) + γ(2/m)R(q, t)where γ = (1/6)(n− 2)/(n − 1) with n = dim(M) and R is a curvaturescalar. Now for a Riemannian geometry ds2 = gik(q)dqidqk it is standardthat in a transplantation qi → qi +δqi one has δAi = Γi

kAdqk, and here

it is assumed that for = (gikAiAk)1/2 one has δ = φkdqk where the φk

are covariant components of an arbitrary vector of M (Weyl geometry).Thus the actual affine connections Γi

k can be found by comparing this

1. RESUME 283

with δ2 = δ(gikAiAk) and one finds

(1.38) Γik = −

i

k

+ gim(gmkφ + gmφk − gkφm)

Thus we may prescribe the metric tensor gik and φi and determine via(1.38) the connection coefficients. Covariant derivatives are defined viacommas and the curvature tensor Ri

km in Weyl geometry is introducedvia Ai

,k, − Ai,,k = F i

mkAm from which arises the standard formula of

Riemannian geometry Rimk = −∂Γi

mk+∂kΓim+Γi

nΓnmk−Γi

nkΓnm where

(1.38) must be used in place of the Riemannian Christoffel symbols. TheRicci symmetric tensor Rik and the scalar curvature R are defined viaRik = R

ik and R = gikRik, while

(1.39) R = R + (n− 1)[(n− 2)φiφi − 2(1/

√g)∂i(

√gφi)]

where R is the Riemannian curvature built by the Christoffel symbols.Now the geometry is to be derived from physical principles so the φi

cannot be arbitrary but are obtained by the same (averaged) least actionprinciple giving the motion of the particle (statistical determination ofgeometry) and when n ≥ 3 the minimization involves only (1.39). Oneshows that ρ(q, t) = ρ(q, t)/

√g transforms as a scalar in a coordinate

change and this will be called the scalar probability density of the randommotion of the particle. Starting from ∂tρ + ∂i(ρvi) = 0 a manifestlycovariant equation for ρ is found to be

(1.40) ∂tρ + (1/√

g)∂i(√

gviρ) = 0

Some calculation then yields a minimum over R when

(1.41) φi(q, t) = −[1/(n− 2)]∂i[log(ρ)(q, t)]

This shows that the geometric properties of space are indeed affectedby the presence of the particle and in turn the alteration of geometryacts on the particle through the quantum force fi = γ(2/m)∂iR whichaccording to (1.39) depends on the gauge vector and its derivatives. It isthis peculiar feedback between the geometry of space and the motion ofthe particle which produces quantum effects. In this spirit one goes nextto a geometrical derivation of the SE. Thus inserting (1.41) into (1.39)one gets

(1.42) R = R + (1/2γ√

ρ)[1/√

g)∂i(√

ggik∂k

√ρ)]

where the value γ = (1/6)[(n− 2)/(n− 1)] has been used. On the otherhand the HJ equation can be written as

(1.43) ∂tS + HC(q,∇S, t)− γ(2/m)R = 0

When (1.42) is introduced into (1.43) the HJ equation and the continuityequation (1.40), with velocity field given by vi = (∂H/∂pi)(q,∇S, t), forma set of two nonlinear PDE which are coupled by the curvature of space.Therefore self consistent random motions of the “particle” are obtainedby solving (1.40) and (1.43) simultaneously. For every pair of solutions


S(q, t, ρ(q, t)) one gets a possible random motion for the particle whoseinvariant probability density is ρ. The present approach is so differentfrom traditional QM that a proof of equivalence is needed and this is onlydone for Hamiltonians of the form HC(q, p, t) = (1/2m)gik(pi−Ai)(pk−Ak) + V (which is not very restrictive) leading to

(1.44) ∂tS +1

2mgik(∂iS −Ai)(∂kS −Ak) + V − γ

2

mR = 0

(R in (1.43)). The continuity equation (1.40) is

(1.45) ∂tρ + (1/m√

g)∂i[ρ√

ggik(∂kS −Ak)] = 0

Owing to (1.42), (1.44) and (1.45) form a set of two nonlinear PDE whichmust be solved for the unknown functions S and ρ. Then a straightfor-ward calculation shows that, setting ψ(q, t) =

√ρ(q, t)exp](i/)S(q, t)]

the quantity ψ obeys a linear SE

(1.46) i∂tψ =1

2m

[i∂i

√g

√g

+ Ai

]gik(i∂k + Ak)

ψ +

[V − γ

2

mR

]ψ

where only the Riemannian curvature R is present (any explicit referenceto the gauge vector φi having disappeared).

We recall that in the nonrelativistic context the quantum potentialhas the form Q = −(2/2m)(∂2√ρ/

√ρ) (ρ ∼ ρ here) and in more di-

mensions this corresponds to Q = −(2/2m)(∆√

ρ/√

ρ). The continuityequation in (1.45) corresponds to ∂tρ + (1/m

√g)∂i[ρ

√ggik(∂kS)] = 0

(ρ ∼ ρ here). For Ak = 0 (1.44) becomes ∂tS + (1/2m)gik∂iS∂kS + V −γ(2/m)R = 0. This leads to an identification Q ∼ −γ(2/m)R whereR is the Ricci scalar in the Weyl geometry (related to the Riemann-ian curvature built on standard Christoffel symbols via (1.39). Hereγ = (1/6)[(n− 2)/(n− 1)] which for n = 3 becomes γ = 1/12; further by(1.41) the Weyl field is φi = −∂ilog(ρ). Consequently for the SE (1.46)in Weyl space the quantum potential is Q = −(2/12m)R where R isthe Weyl-Ricci scalar curvature. For Riemannian flat space R = 0 thisbecomes via (1.42)

(1.47) R =1

2γ√

ρ∂ig

ik∂k√

ρ ∼ 12γ

∆√

ρ√

ρ⇒ Q = − 2

2m

∆√

ρ√

ρ

as desired; the SE (1.46) reduces to the standard SE i∂tψ = −(2/2m)∆ψ+V ψ (Ak = 0). Moreover (1.39) provides an interaction between gravity(involving R and g) and QM (which generates φi via ρ and R via Q).

(7) In [841] the KG equation is also derived via an average action principlewith the restriction of a priori Weyl geometry removed. The spacetimegeometry is then obtained from the average action principle to be ofWeyl type with a gauge field φµ = ∂µlog(ρ). One has a kind of “moral”equivalence between QM in Riemannian spaces and classical statisticalmechanics in a Weyl space. Traditional QM based on wave equations andad hoc probability calculus is merely a convenient tool to overcome thecomplications arisin from a nontrivial spacetime geometrical structure.

1. RESUME 285

In the KG situation there is a relation m2 − (R/6) ∼ M2 ∼ m2(1 + Q)(approximating m2exp(Q)) and Q ∼

√ρ/m2√ρ which implies R/6 ∼

−√

ρ/√

ρ.(8) Referring to Item 3 in Section 6.1.1 we note that for φµ ∼ Aµ = ∂µlog(P ) (P ∼

ρ) one can envision a complex velocity pµ + iλAµ leading to

(1.48) |pµ + i√

λAµ|2 = p2µ + λA2

µ ∼ gµν

(∂S

∂xµ

∂S

∂xν+

λ

P 2

∂P

∂xµ

∂P

∂xν

)This is exactly the term arising in a Fisher information Lagrangian

(1.49)


P

∂S

∂t+

12gµν

[∂S

∂xµ

∂S

∂xν+

λ

P 2

∂P

∂xµ

∂P

∂xν

]+ V

dtdnx

where I is the information term (see Section 3.1)


2

∫1P

∂P

∂yi

∂P

∂ykdny

known from φµ. Hence we have a direct connection between Fisher in-formation and the Weyl field φµ along with a motivation for a complexvelocity (cf. [223]). Further we note, via [189] and quantum geometryin the form ds2 ∼

∑dp2

j/pj on a space of probability distributions, that(1.50) can be defined as a Fisher information metric (positive definite viaits connection to (∆N)2) and

(1.51) Q ∼ −22gµν

[1

P 2

∂P

∂xµ

∂P

∂xν− 2

P

∂2P

∂xµ∂xν

](corresponding to −(2/2m)(∂2√ρ/

√ρ) = −(2/8m)[(2ρ′′/ρ)−(ρ′/ρ)2]).

Further from u = −Dφ with Q = D2((1/2)|u|2−∇ · φ), one expresses Qdirectly in terms of the Weyl vector. This enforces the idea that QM isbuilt into Weyl geometry and moreover that fluctuations generate Weylgeometry.

(9) In Section 3.3 the WDW equation is treated following [876, 870] froma Bohmian point of view. One builds up a Lagrangian and Hamiltonianin terms of lapse and shift functions with a quantum potential

(1.52) Q =∫

d3xQ; Q = 2NqGijk1|ψ|

δ2|ψ|δqijδk

The quantum potential changes the Hamiltonian constraint algebra torequire weak closure (i.e. closure modulo the equations of motion); reg-ularization and ordering are not considered here but will not affect theconstraint algebra. The quantum Einstein equations are derived in theform

(1.53) Gij = − 1

N

δQ

δqij; G

0µ =Q

2√−g

g0µ

The Bohmian HJ equation is

(1.54) GijkδS

δqij

δS

δqk−√q (3R−Q) = 0

where S is the phase of the WDW wave function and this leads to thesame equations of motion (1.53). The modified Einstein equations aregiven in Bohmian form via

(1.55) Gij = −κT

ij − 1N

δ(QG + Qm)δgij

; G0µ = −κT

0µ +QG + Qm

2√−g

g0µ

(1.56) Qm = 2 N√

q

21|ψ|

δ2|ψ|δφ2

; QG = 2NqGijk1|ψ|

δ2|ψ|δqij ; δqk

QG =∫

d3xQG; Qm =∫

d3xQm

In the third paper of [876] the Ashtekar variables are employed and itis shown that the Poisson bracket of the Hamiltonian with itself changeswith respect to its classical counterpart but is still weakly equal to zero(modulo regularization, etc.); we refer to [876] and Section 4.3.1 fordetails.

1.3. GEOMETRY, GRAVITY, AND QM. We have already indicatedsome interaction of QM and geometry via Bohmian mechanics and remark hereupon other aspects.

(1) It is known that one can develop a quantum geometry via Kahler geom-etry on a preHilbert space P (H) (see e.g. [54, 153, 188, 189, 203,244, 245, 246, 247, 248]). Thus P (H) is a Kahler manifold with aFubini-Study metric based on (|dψ⊥ >= |dψ > −|ψ >< ψ|dψ >)

(1.57)14ds2

PS = [cos−1(| < ψ|ψ > |)]2 ∼ 1− | < ψ|ψ > |2 =< dψ⊥|dψ⊥ >

where ds2PS =

∑dp2

j/pj =∑

pj(d log(pj))2 gives the connection to prob-ability distributions. We have already seen in Item 8 of Section 6.1.2 howthis probability metric is related to Fisher information, fluctuations, andthe quantum potential.

(2) There is a fascinating series of papers by Arias, Bonal, Cardenas, Gonza-lez, Leyva, Martin, and Quiros dealing with general relativity (GR) andconformal variations (cf. Section 3.2.2). We omit details here but simplyremark that conformal GR with gab = Ω2gab is shown to be the only con-sistent formulation of gravity. Here consistent refers to invariance underthe group of transformations of units of length, time, and mass.

(3) In Section 4.5.1 one goes into the Bohmian interpretation of quantumcosmology a la [770, 772, 774, 961] for example (cf. also [123, 124,571, 572, 573]). Thus write H =

∫d3x(NH + N jHj) where for GR

with a scalar field

(1.58) Hj = −2Diπijπφ∂jφ; H = κGijkπ

ijπk +12h−1/2π2

φ+

+h1/2

[−κ−1(R(3) − 2Λ) +


]The canonical momentum is

(1.59) πij = −h1/2(Kij − hijK) = Gijk(hk −DkN −DNk);

1. RESUME 287

Kij = − 12N

(hij −DiNj −DjNi)

K is the extrinsic curvature of the 3-D hypersurface Σ in question withindices lowered and raised via the surface metric hij and its inverse) andπφ = (h1/2/N)(φ−N j∂jφ) is the momentum of the scalar field (Di is thecovariant derivative on Σ). Recall also the deWitt metric

(1.60) Gijk =12h−1/2(hikhj + hihjk − hijhk)

The classical 4-metric and scalar field which satisfy the Einstein equationscan be obtained from the Hamiltonian equations

(1.61) ds2 = −(N2 −N iNi)dt2 + 2Nidxidt + hijdxidxj ;

hij = hij ,H; πij = πij ,H; φ = φ,H; πφ = πφ,HOne has the standard constraint equations which when put in Bohmianform with ψ = Aexp(iS/) become

(1.62)

−2hiDjδS(hijφ)

δhj+

δS(hij , φ)δφ

∂iφ = 0; −2hiDjδA(hij , φ)

δhj+

δA(hij , φ)δφ

= 0

These depend on the factor ordering but in any case will have the form

(1.63) κGijkδS

δhij

δS

δhk+

12h−1/2

(δS

δφ

)2

+ V + Q = 0

Q = −2

A

(κGijk

δ2A

δhijδhk+

h−1/2

2δ2A

δφ2

)

(1.64) κGijkδ

δhij

(A2 δS

δhk

)+

12h−1/2 δ

δφ

(A2 δS

δφ

)= 0

Now in the dBB interpretation one has guidance relations

(1.65) πij =δS(hab, φ)

δhij; πφ =

δS(hij , φ)δφ

One then develops the Bohmian theory and from (1.65) results

(1.66) hij = 2NGijkδS

δhk+ DiNj + DjNi; φ = Nh−1/2 δS

δφ+ N i∂iφ

The question posed now is to find what kind of structure arises from(1.66). The Hamiltonian is evidently HQ =

∫d3x

[N(H + Q) + N iHi

];

HQ = H + Q and the first question is whether the evolution of the fieldsdriven by HQ forms a 4-geometry as in classical gravitational dynamics.Various situations are examined and (for Q of a specific form) some-times the quantum geometry is consistent (i.e. independent of the choiceof lapse and shift functions) and forms a nondegenerate 4-geometry (ofEuclidean type). However it can also be consistent and not form a non-degenerate 4-geometry. In general, and always when the quantum po-tential is nonlocal, spacetime is broken and the evolving 3-geometries do


not stick together to form a nondegenerate 4-geometry. These are veryinteresting results and mandate further study.

(4) Next (cf. Section 4.6) one goes to noncommutative (NC) theories folow-ing [77, 78, 79, 772]. First from [77, 78] one considers canonical co-mutation relations [Xµ, Xν ] = iθµν and develops a Bohmian theory fora noncommutative QM (NCQM) via a Moyal product

(1.67) (f ∗ g) =1

(2π)n

∫dmkdnpei(kµ+pµ)xµ−(1/2)kµθµνpν f(k)g(p) =

For θ0i = 0 one has a Hilbert space as in commutative QM with a NCSE

(1.68) i∂ψ(xi, t)

∂t= − 2

2m∇2ψ(xi, t) + V (xi) ∗ ψ(xi, t) =

=2

2m∇2ψ(xi, t) + V

(xj + i

θjk

2∂k

)ψ(xi, t)

The operators Xj = xj + iθjk∂k

2 are the observables with canonical co-ordinates xi and ρd3x = |ψ|2d3x is intepreted as the porbability thatthe system is in a region of volume d3x around x at time t. One writesψ = Rexp(iS/) and there results

(1.69)∂S

∂t+

(∇S)2

2m+V +Vnc +QK +QI = 0; Vnc = V

(xi − θij

2∂jS

)−V (xi);

QK = (− 2

2m

∇2ψ

ψ

)−(

2

2m(∇S)2

)= − 2

2m

∇2R

R;

QI = (

V [xj + (iθjk/2)∂k]ψψ

)− V

(xi − θij

2∂jS

)One arrives at a formal structure involving

(1.70) Xj = xi + iθjk∂k/2; Xi(t) = xi(t)− (θij/2)∂jS(xi(t), t);

dxi(t)dt

=[∂iS(x, t)

m+

θij

2

∂V (Xk)∂Xj

+Qi

2

]∣∣∣∣xi=xi(t)

One finds then

(1.71)∂ρ

∂t+

∂(ρxi

∂xi− ∂

∂xi

[ρ

(θij

2

∂V (Xk)∂Xj

+Qi

2

)]+ Σθ = 0

so for equivariance (ρ = |ψ|2) it is necessary that the sum of the lasttwo terms in (1.71) vanish and when V (Xi) is linear or quadratic thisholds. In [77, 78] one also looks at NC theory in Kantowski-Sachs (KS)universes and at Friedman-Robertson-Walker (FRW) universes with aconformally coupled scalar field. Many specific situations are examined,especially in a minisuperspace context.

1. RESUME 289

(5) Next (cf. Item 3 in Section 6.1.1 and Item 8 in Section 6.1.2) we con-sider [449, 444]. One gives a new derivation of the SE via the exactuncertainty principle and a formula

(1.72) Hq[P, S] = Hc[P, S] + C

∫dx∇P · ∇P

2mP

for the quantum situation. Consider then the gravitational framework

(1.73) ds2 = −(N2 − hijNiNj)dt2 + 2Nidxidt + hijdxidxj

One introduces fluctuations via πij = (δS/δhij) + f ij and arrives at aWDW equation

(1.74)[−2

2δ

δhijGijk

δ

δhk+ V

]Ψ = 0

Note that an operator ordering is implicit and thus ordering ambiguitiesdo not arise (similarly for quantum particle motion). The work here in[449, 444] is significant and very interesting; it is developed in somedetail in Section 4.7.

(6) In Section 4.1 we followed work of M. Israelit and N. Rosen on Dirac-Weyl geometry (see in particular [498, 499, 817]). Recall that in Weylgeometry gµν → gµν = exp(2λ)gµν is a gauge transformation and for avector B of length B one has dB = Bwνdxν where wν ∼ φν is the Weylvector. The Weyl connection coefficients are

(1.75) ∆Bλ = BσKλσµνdxµδxν ; ∆B = BWµνdxµδxν

and under a gauge transformation wµ → wµ = wµ + ∂µλ. One writesWµν = wµ,ν − wν,µ (where commas denote partial derivatives). TheDirac-Weyl action here is given via a field β (β → β = exp(−λ)β undera gauge transformation) in the form

(1.76) I =∫



Note here the difference in appearance from (1.37) or the Dirac formin Appendix D; these are all equivalent after suitable adjustment (cf.Remark 6.1). Under parallel transport ∆B = BWµνdxµδxν so onetakes Wµν = 0 via wν = ∂νw and we have what is called an inte-grable Weyl geometry with generating elements (gµν , w, β). Further setbµ = ∂µ(log(β)) = βµ/β and use a modified Weyl connection vectorWµ = wµ + bµ. Then varying (1.76) in w and gµν gives

(1.77) 2(κβ2W ν);ν = S; Gνµ = −8π

T νµ

β2+

+16πκ

(W νWµ −

12δνµW σWσ

)+ 2(δν

µbσ;σ − bν

;µ) + 2bνbµ + δνµbσ

σ − δνµβ2Λ


where S is the Weyl scalar charge 16πS = δLM/δw, Gνµ is the Einstein

tensor, and the energy momentum tensor of ordinary matter is

(1.78) 8π√−gTµν = δ(

√−gLM )/δgµν

Finally variation in β gives an equation for he β field

(1.79) R + k(bσ;σ + bσbσ) = 16πκ(wσwσ − wσ

;σ) + 4β2Λ + 8πβ−1B

(here 16πB = δLM/δβ is the Dirac charge conjugate to β). Note

(1.80) δIM = 8π∫

(Tµνδgµν + 2Sδw + 2Bδβ)√−gd4x

yielding the energy momentum relation Tλµ;λ−Swµ−βBbµ = 0. Actually

via (1.77) with S + T = βB one obtains again (1.79) which is seentherefore as a corollary and not an independent equation. One derivesnow conservation laws etc. and following [817] produces an equation ofmotion for a test particle. Thus consider matter made up on identicalparticles of rest mass m and Weyl scalar charge qs, being in the stage ofa pressureless gas so Tµν = ρUµUν where Uν is the 4-velocity and notealso Tλ

µ;λ − Tbµ = SWµ. Then one arrives at

(1.81)dUµ

ds+

µλ σ

UλUσ =

(bλ +

qs

mWλ

)(gµλ − UµUλ)

Further a number of illustrations are worked out involving the creation ofmass like objects from Weyl-Dirac geometry, in a FRW universe for ex-ample (i.e. an external observer sitting in Riemannian spacetime wouldrecognize the object as massive). Cosmological models are also con-structed with the Weyl field serving to create matter. The treatment isextensive and profound.

REMARK 7.1.1. We have encountered Dirac-Weyl-Bohm (DWB) in Section2.1 (Section 7.1.2, Item 5) and Dirac-Weyl geometry in Section 4.1 and AppendixD (Section 7.1.3, Item 6). The formulations are somewhat different and we trynow to compare certain features. In Section 7.1.2 one has

(1.82) A =∫

d4x√−g(FµνFµν − β2 WR+ (σ + 6)β;µβ;µ + Lmatter

for the action; in Section 7.1.3 the action is

(1.83) I =∫



and in Appendix D we have (for the simplest vacuum equations and g =√−g)

(1.84) I =∫

[(1/4)FµνFµν − β2R + 6βµβµ + cβ4]gd4x

We recall the idea of co-covariant derivative from Appendix D where for a scalarS of Weyl power n one has S∗µ = S;µ = nwµS ≡ Sµ − nwµS (Sµ = ∂µS) and I in

1. RESUME 291

(1.84) is originally

(1.85) I =∫

[(1/4)FµνFµν − β2∗R + kβ∗µβ∗µ + cβ4]gd4x

However β is a co-scalar of weight −1 and β∗µβ∗µ = (βµ + βwµ)(βµ + βwµ) sousing

(1.86) −β2∗R + kβ∗µβ∗µ = −β2R + kβµβµ + (k − 6)β2κµκµ+

+6(β2κµ):µ + (2k − 12)βκµβµ

one obtains (1.84) for k = 6. Further recall that Wµ = wµ+∂µlog(β) so Wµν = Fµν

and c in (1.84) corresponds to Λ in (1.83). The notation WR in (1.82) is the sameas R and β;µ ∼ βµ = ∂µβ so (σ+6) = 6 provides a complete identification (modulomatter terms to be added in (1.84)); note σ = k−6 from [817]. Now in [872, 873]one takes a Dirac-Weyl action of the form (1.82) and relates it to a Bohmian theoryas in Section 3.2.1. The same arguments hold also for σ = 0 here with

(1.87) β ∼M;8πT

R∼ m2; α =

2

m2c2∼ − 6

R; ∇νT

µν − T∇µβ

β= 0

Note also 16πT = βψ where ψ = δLM/δβ ∼ 16πB so B ∼ T/β. One assumeshere that 16πJµ = δLM/δwµ = 0 where wµ ∼ φµ.

REMARK 7.1.2. We recall from Section 3.2 that β ∼ M and M2/m2 =exp(Q) is a conformal factor. Further for β0 → β0exp(−Ξ(x)) one has wµ →wµ + ∂µΞ where −Ξ = log(β/β0) showing an interplay between mass and geom-etry. Recall also the relation ∇µ(βwµ + β∇µβ) = 0. This indicates a number ofconnections between the quantum potential, geometry, and mass. Hence virtuallyany results in Dirac-Weyl theory models will involve the quantum potential. Thisis made explicit in Section 3.2 from [873] and could be developed for the examplesand theory from [499] once wave functions and Bohmian ideas are inserted.

1.4. GEOMETRIC PHASES. We go now to [283] for some remarks ongeometric phase and the quantum potential. One refers back here to geometricphases of Berry [108] and Levy-Leblond [603] for example where the latter showsthat when a quanton propagates through a tube, within which it is confined byimpenetrable walls, it acquires a phase when it comes out of the tube. Thusconsider a tube with square section of side a and length L. Before entering the tubethe quanton’s wave function is φ = exp(ipx/) where p is the initial momentum.In the tube the wave function has the form

(1.88) ψ = Sin(nxπ

x

a

)Sin

(nyπ

y

a

)exp(ip′x/)

with appropriate transverse boundary conditions. After entering the tube theenergy E of the quanton is unchanged but satisfies

(1.89) E =(p′)2

2m(n2

x + n2y)

π2

2ma

For the simplest case nx = ny = 1 it was found that after the quanton left thetube there was an additional phase

(1.90) ∆Φ =π22

pa2L

Subsequently Kastner [539] related this to the quantum potential that arises inthe tube. Thus let the wave function in the tube be Rexp[(iS/)+(ipx/)] in polarform. The eventual changes in the phase of the wave function, due to the tube,are now concentrated in S. In order to single out the influence of the tube on thewave function write ψ1 = ψexp(ipx/) and the quantum potential correspondingto ψ is then

(1.91) Q = − 2

2m

∆R

R=

π22

ma2

Now turn to the laws of parallel transport where for the Berry phase the law ofparallel transport for the wave function is (cf. [892])

(1.92) < ψ|ψ >= 0

For the Levy-Leblond phase the law of parallel transport is given by

(1.93) < ψ|ψ >= −1Q|ψ|2

In this approach the wave function acquires an additional phase after the quantonhas left the tube in the form

(1.94) ψ(t + ∆t) = exp(−iQ∆t/)ψ(t)

which after expansion in ∆t leads to the law of parallel transport in (1.93). Indeed

(1.95) Q∆t =π22

ma2∆t =

π22

ma2

mL

p=

π22

pa2L = ∆Φ

If we use the polar form for the wave function (1.93) gives (∂S/∂t) = −Q and thismeans that this new law of parallel transport eliminates the quantum potentialfrom the quantum HJ equation. The whole quantum information is now carried bythe phase of the wave function. One can see that the nature of this phase is quitedifferent from Berry’s phase; it is related to the presence or not of constraints inthe system (in this case the tube).

Now consider a quite different type of constrained system where again a newgeometric phase will arise. Look at a quantum particle constrained to move on acircle. The wave function has the form ψ ∼ sin(ns/ρ0) where ρ0 is the radius ofthe circle and s is the arc length with origin at a tangent point. Then the wavefunction will have a node at this tangent point. For a circle the value n = 1/2 isalso allowed (cf. [467, 570]) and the corresponding quantum potential for n = 1/2is

(1.96) Q =h2

8mρ20

which is exactly equal to the constant E0 appearing in the Hamiltonian for aparticle on a circle with radius ρ0 following the Dirac quantization procedure

2. HYDRODYNAMICS AND GEOMETRY 293

for constrained systems (cf. [846]). The phase which a quanton would acquiretraveling along the circle is then

(1.97) Q2πρ0m

p=

π2

4ρ0p

Note that if the circle becomes very small then the geometrical phase can not getbigger than ∼ (/m). This limit is imposed by the Heisenberg uncertainty relationρ0p ∼ . This is not the case for the Levy-Leblond phase which can get very largeprovided L >> a.

1.5. ENTROPY AND CHAOS. Connections of the quantum potential toFisher information have already been recalled in e.g. Section 7.1.1 and we recallhere from Chapter 6 a few matters.

(1) An extensive discussion relating Fisher information (as a “mother” infor-mation) to various forms of entropy is developed in Chapter 6 and thisgives implicitly at least many relations between entropy, kinetic theory,uncertainty, and the quantum potential.

(2) A particular result of interest in Section 6.2.1 shows how the quantumpotential acts as a constraining force to prevent deterministic chaos.

2. HYDRODYNAMICS AND GEOMETRY

We mentioned briefly some hydrodynamical aspects of the SE in Sections 1.1and 1.3.2 and return to that now following [294]. Here one wants to limit the roleof statistics and measurement to unveil some geometric features of the so calledMadelung approach. Thus, with some repetition from Section 1.1, consider a SE(/i)ψt + H(x, (/i)∇)ψ = 0 with ψ = Rexp(iS/) to arrive at

(2.1)∂S

∂t+ H(x,∇S)− 2

2m

∆R

R= 0;

∂P

∂t+

∂

∂xi(Pxi) = 0; xi =

[∂H

∂pi

]p=∇S

(where P = R2), and Madelung equations of the form (cf. (1.5))

(2.2)∂S

∂t+ H(x,∇S)− 2

2m

∆√

ρ√

ρ= 0;

∂ρ

∂t+

∂

∂xi(ρxi) = 0

where, in a continuum picture, ρ = mP is the mass density of an extended particlewhose shape is dictated by P. Setting vi = xi one has then an Euler equation ofthe form

(2.3)∂

∂t(ρvi) +

∂

∂xk(ρvivk) = − ρ

m

∂V

∂xi+

∂

∂xkτ ik

Following [924] one has expressed the quantum force term here as the divergenceof a symmetric “quantum stress” tensor

(2.4) τij =(

2m

)2

ρ∂2(log(ρ))

∂xi∂xj; pi =

2

2m2ρ

∂2σ

∂(xi)2

where pi denotes diagonal elements or principal stresses expressed in normal co-ordinates (with ρ = exp(2σ)). The stress pi is tension like (resp. pressure like) if

pi > 0 (resp. pi < 0) and the mean pressure is

(2.5) p = −13Tr(τij) = − 2

6m2ρ∆σ

In classical hydrodynamics negative pressures are often associated with cavitationwhich involves the formation of topological defects in the form of bubbles. Foran ideal fluid one would need τij = −pδoj and this occurs if and only if the massdensity is Gaussian σ ∝ −xixi in which case p ∝ (2/2m)ρ. Generally the stresstensor will not be isotropic, and not an ideal fluid; moreover if one had a viscousfluid one would expect τij to be coupled to the rate of deformation tensor (derivedfrom Dv). Since this does not occur one does not call this form of matter a fluidbut rather a Madelung continuum, corresponding to something like an inviscidfluid which also supports shear stresses, whereas the Gaussian wave packet of QMcorresponds to an ideal compressible irrotational fluid medium.

If now one adds time as the zeroth coordinate and extends the velocity vectorby v0 = 1 then, defining the energy momentum tensor as

(2.6) Tµν = ρ

[vµvν −

(

2m

)2∂2(log(ρ))∂xµ∂xν

]then the Euler and continuity equations can be combined in the form ∂Tµν/∂xµ =−(ρ/m)∂νV ; this is somewhat misleading since it is based on a nonrelativisticapproach but it leads now to the relativistic theory. First start with the KGequation

(2.7)[−2ηµν ∂2

∂xµ∂xν+ m2

0c2

]ψ = 0

For ψ = Rexp(iS/) one gets now

(2.8) ηµν ∂S

∂xµ

∂S

∂xν+ m2

0c2 − 2 R

R= 0; ηµν ∂

∂xµ

(P

∂S

∂xν

)= 0

Define now the 4-velocity, rest mass energy, energy momentum, and stress tensorvia

(2.9) uµ =1

m0

∂S

∂xµ; ρ = m0P ; pµ = ρuµ;

τµν =(

2m0

)2∂2(log(ρ))∂xµ∂xν

; Tµν = ρ[uµuν + τµν ]

to arrive at relativistic equations for the medium described by ρ and uµ in theform (♣) ∂µTµν = 0 and ∂µpµ = 0 (the second equation is an incompressibilityequation and this does not contradict the nonrelativistic compressibility of themedium since in relativity incompressibility in fluid media is equivalant to an in-finite speed of light corresponding to rigidity in solid media).

One then erects an elegant mathematical framework involving spacetime foli-ations related to the complex character of ψ (see also the second paper in [294] onfoliated cobordism, etc.). This is lovely but rather too abstract for the style of this


book so we will not try to reproduce it here; we can however skip to some calcula-tions involving the geometric origin of the quantum potential. Thus consider theconsequences of choosing a scale of unit norm via the function

√ρ. One takes a

conformally related metric g = Ω2g on a manifold M (where Ω2 > 0) and, writingΩ = exp(σ), one obtains the following formulas for the Levi-Civita connection,Ricci curvature, and scalar curvature

(2.10) Γijk = Γi

jk = δij∂kσ + δi

k∂jσ − gjkgi∂σ;

Rij = Rij − (n− 2)σij − [∆σ + (n− 2)(∂kσ∂kσ)]gij

R = e−2σ[R− 2(n− 1)∆σ − (n− 1)(n− 2)(∂iσ∂iσ)]

where σij = ∂i∂jσ − ∂jσ∂iσ. Now if the constant m is replaced by the function ρthen one must contend with the derivatives ∂iρ and for ρ = exp(2σ) the Minkowskimetric will be deformed from g = η, Γi

jk = 0, Rij = R = 0 to

(2.11) Γijk = δi

j∂kσ + δik∂jσ − ηjkηi∂σ; Rij = −2σij − [σ + 2(∂kσ∂kσ)]ηij

R = −6e−2σ[σ + (∂iσ∂iσ]

Putting Ω back into the equation for scalar curvature one obtains

(2.12) R = − 6Ω2

(ΩΩ

);

√

ρ√

ρ=

ΩΩ

= −16RΩ2 = −1

6ρR

This identifies the quantum potential as a mass density times a scalar curvatureand resembles some results obtained earlier from [840, 841] for example (cf. Sec-tion 3.3 and 3.3.2). One has also

(2.13) ∂iRij = ∂i

(12gijR

)=

12∂jR ⇒ ∂iRij = −3∂iτij

so the Takabayashi stress tensor differs from the Ricci curvature only by a termof vanishing divergence. Hence there is no loss of generality in using the Riccicurvature of g = ρη as the stress tensor since both define the same force field;this means in particular that one is dealing with principal curvatures instead ofprincipal stresses. To extend all this to a more general Lorentz manifold onenotes that under a conformal change of spacetime metric to the energy metric theEinstein tensor becomes

(2.14) Gµν = Rµν −12gµνR = Gµν − 2σµν + [2σ + ∂λσ∂λσ]gµν

Thus it seems to assume that this implies a quantum correction to the Einsteinequation

(2.15) Gµν +(

σ +12∂λσ∂λσ

)gµν = 8πGTµν + 2σµν


2.1. PARTICLE AND WAVE PICTURES. An interesting discussion ofhydrodynamic features of QM, electrodynamics, and Bohmian mechanics appearsin [474] and we will sketch more or less thoroughly a few ideas here. Thus a hydro-dynamic model of QM provides an interpretation of two pictures, wave mechanical(Eulerian) and particle (Lagrangian), and the two versions of QM have associatedHamiltonian formulations that are connected by a canonical transformation. Thisgives a new and precise meaning to the notion of wave-particle duality. Howeverit is necessary to distinguish the dBB corpuscle from a fluid particle. Considera fluid as a continuum of particles with history encoded in the position variablesq(a, t) where each particle is distinguished by a continuous vector label a. Themotion is continuous in that the mapping from a-space to q-space is single valuedand differentiable with inverse a(q, t). Let ρ0(a) be the initial quantum probabilitydensity with

∫ρ0(a)d3a = 1. Introduce a mass parameter m so that the mass of

an elementary volume d3a attached to the point a is given by mρ0(a)d3a. Note∫mρ0(a)d3a = m so this is a total mass of the system. The conservation of mass

of a fluid element in the course of its motion is

(2.16) mρ(q(a, t))d3q(a, t) = mρ0(a)d3a; ρ(a, t) = J−1(a, t)ρ0(a);

J =13!

εijkεmn∂qi

∂a

∂qj

∂am

∂qk

∂an

J is the Jacobian of the transformation between the two sets of coordinates andεijk is the completely antisymmetric tensor with ε123 = 1. Let V be the potentialof an external classical body force and U the internal potential energy of the fluiddue to interparticle interactions. Here assume U depends on ρ(q) and its firstderivatives and hence via (2.16) on the second order derivatives of q with respectto a. The Lagrangian is then (with integrand = [· · · ])

(2.17) L[q, qt, t] =∫ [

12mρ0(a)

(∂q(a, t)

∂t

)2

− ρ0(a)U(ρ)− ρ0(a)V (q(a))

]d3a

(one substitutes for ρ from (2.16)). It is the action of the conservative forcederived from U on the trajectories that represents the quantum effects here. Theyare characterized by the following choice for U, motivated by the known Eulerianexpression for internal energy,

(2.18) U =2

8m

1ρ2

∂ρ

∂qi

∂ρ

∂qi=

2

8m

1ρ20

JijJik∂

∂aj

(ρ0

J

) ∂

∂ak

(ρ0

J

);

∂

∂qi= J−1Jij

∂

∂aj; Ji =

∂J

∂(∂qi/∂a)=

12εijkεmn

∂qj

∂am

∂qk

∂an;

∂qk

∂ajJki = Jδij

Thus Ji is the cofactor of ∂qi/∂a. The interaction in the quantum case is notconceptually different from classical fluid dynamics but differs in that the order ofderivative coupling of the particles is higher than in a classical equation of state.The Euler-Lagrange equations for the coordinates are

(2.19)∂

∂t

∂L

∂(∂qi(a, t)/∂t)− δL

δqi(a)= 0;


δL

δqi=

∂

∂qi− ∂

∂aj

∂

∂(∂qi/∂aj)+

∂2

∂aj∂ak

∂

∂(∂2qi/∂aj∂ak)which yield

(2.20) mρ0(a)∂2qi(a)

∂t2= −ρ0(a)

∂V

∂qi− ∂Wij

∂aj;

Wij = −ρ0(a)∂U

∂(∂qi/∂aj)+

∂

∂ak

(ρ0(a)

∂U

∂(∂2qi/∂aj∂ak)

)This has the form of Newton’s second law and instead of giving an explicit formfor Wij one uses a more useful tensor σij defined via Wik = Jjkσij where σij isthe analogue of the classical pressure tensor pδij . Using (2.18) one can invert toobtain(2.21)

σij = J−1Wik∂qj

∂ak=

2

4mJ3Jik

[ρ−10 Jj

∂ρ0

∂ak

∂ρ0

∂a+ (J−1JjJmn − JjmJn)×

×∂ρ0

∂a

∂2qm

∂ak∂an− Jj

∂2ρ0

∂ak∂a+ ρ0(J−1JmnJjrs + J−1JjJmrns − 2J−2JjJmnJrs)×

× ∂2qr

∂ak∂as

∂2qm

∂a∂an+ ρ0J

−1JjJmn∂3qm

∂ak∂a∂an

];

Jjmn =∂Jj

∂(∂qm/∂an)= εjmkεnr

∂qk

∂ar

One checks that σij is symmetric and the equation of motion of the ath particlemoving in the field of the other particles and the external force is then

(2.22) mρ0(a)∂2qi(a)

∂t2= −ρ0(a)

∂V

∂qi− Jkj

∂σik

∂aj;

∂Jij

∂aj= 0

(the latter equation is an identity used in the calculation). The result in (2.22) isthe principal equation for the quantum Lagrangian method; its solutions, subjectto specification of ∂qi0/∂t, lead to solutions of the SE. Multiplying by ∂qi/∂ak oneobtains the Lagrangian form

(2.23) mρ0(a)∂2qi(a)

∂t2∂qi

∂ak= −ρ0(a)

∂V

∂ak− ∂qi

∂akJkj

∂σik

∂aj;

pi(a) =∂L

∂(∂qi(a)/∂t)= mρ0(a)

∂qi(a)∂t

A Hamiltonian form can also be obtained via the canonical field momenta pi(a) =∂L/∂(∂qi(a)/∂t) = mρ0(a)(∂qi(a)/∂t) with(2.24)

H =∫

pi(a)∂qi(a)

∂td3a− L =

∫ [p(a)3

2mρ0(a)+ ρ0(a)U(J−1ρ0) + ρ0(a)V (q(a))

]d3a

Hamilton’s equations via Poisson brackets qi(a), qj(a′) = pi(a), pj(a′) = 0 andqi(a), pj(a′) = δij(a − a′) are ∂tqi(a) = δH/δpi(a) and ∂tpi(a) = −δH/δqi(a)which when combined reproduce (2.22). Now to obtain a flow that is representativeof QM one restricts the initial conditions for (2.22) to something corresponding to


quasi-potential flow which means () ∂qi0/∂t = (1/m)(∂S0(a)/∂ai). However theflow is not irrotational everywhere because the potential S0(a) obeys the quanti-zation condition

(2.25)∮

C

∂qi0(a)∂t

dai =∮

C

1m

∂S0(a)∂ai

dai =n

m(n ∈ Z)

where C is a closed curve composed of material paricles. If it exists vorticity occursin nodal regions (where the density vanishes) and it is assumed that C passesthrough a region of good fluid where ρ0 = 0. To show that these assumptionsimply motion characteristic of QM one demonstrates that they are preserved bythe dynamical system. One first puts (2.23) into a more convenient form. Thus,using (2.16), the stress tensor (2.21) takes a simpler form

(2.26) σij =2

4m

(1ρ

∂ρ

∂qi

∂ρ

∂qj− ∂2ρ

∂qi∂qj

)Using

(2.27)1ρ

∂σij

∂qj=

∂VQ

∂qi; VQ =

2

4mρ

(12ρ

∂ρ

∂qi

∂ρ

∂qi− ∂2ρ

∂qi∂qi

)(note VQ is the dBB quantum potential) one sees that (2.23) can also be simplifiedas

(2.28) m∂2qi

∂t2∂qi

∂ak=

∂

∂ak(V + VQ)

Now integrate this equation between limits (0, t) to get

(2.29) m∂qi

∂t

∂qi

∂ak= m

∂qi0

∂t+

∂χ(a, t)∂ak

; χ =∫ t

0

(12m

(∂q

∂t

)2

− V − VQ

)dt

Then using () one has

(2.30)∂qi

∂t

∂qi

∂ak=

1m

∂S

∂ak; S(a, t) = S0(a) + χ(a, t)

with initial conditions q = a, χ0 = 0. To obtain the q-components multiply byJ−1Jik and use (2.18) to get (♠) ∂qi/∂t = (1/m)(∂S/∂qi) where S = S(a(q, t), t).Thus the velocity is a gradient for all time and (♠) is a form of the law of motion.Correspondingly one can write (2.28) as

(2.31) m∂2qi

∂t2= − ∂

∂qi(V + VQ)

This puts the fluid dynamical law of motion (2.22) in a form of Newton’s lawfor a particle of mass m. Note that the motion is quasi potential since the value(2.25) is preserved, i.e. (•) ∂t

∮C(t)

(∂qi/∂t)dqi = 0 where C(t) is the evolute of thematerial particles conposing C (cf. [113]). To obtain the equation governing Suse the chain rule Ft|a = ∂tF |q +(∂tqi)(∂F/∂qi) and since χ = S−S0 from (2.30)one has

(2.32)∂χ

∂t

∣∣∣∣a

=∂S

∂t

∣∣∣∣q

+∂qi

∂t

∂S

∂qi−(

∂S0

∂t

∣∣∣∣q

+∂qi

∂t

∂S0

∂qi

)


The two terms in the bracket sum to ∂S0(a)/∂t = 0 and using (♠) one obtains(∂χ/∂t)|a = (∂S/∂t)|q + m(∂q/∂t)2. Hence from (2.29) and (♠) one has

(2.33)∂χ

∂t

∣∣∣∣a

=12m

(∂q

∂t

)2

− V − VQ ⇒∂S

∂t+

12m

(∂S

∂q

)2

+ V + VQ = 0

This is the quantum HJ equation and one has shown that the equations (2.16),(2.22), and () are equivalent to the 5 equations (2.16), (2.30), and (2.33); theydetermine the functions (qi, ρ, S). Note that although the particle velocity isorthogonal to a moving surface S = c the surface does not keep step with theparticles that initially compose it and hence is not a material surface. There isalso some interesting discussion about vortex lines for which we refer to [474].

The fundamental link between the particle (Lagrangian) and wave mechanical(Eulerian) pictures is defined by the following expression for the Eulerian density

(2.34) ρ(x, t) =∫

δ(x− q(a, t))ρ0(a)d3a

The corresponding formula for the Eulerian velocity is contained in the expressionfor the current

(2.35) ρ(x, t)vi(x, t) =∫

∂qi(a, t)∂t

δ(x− q(a, t))ρ0(a)d3a

These relations play an analogous role in the approach to the Huygen’s formula

(2.36) ψ(x, t) =∫

G(x, t; a, 0)ψ0(a)d3a

in the Feynman theory; thus one refers to δ(x−q9a, t)) as a propagator. Unlike themany to one mapping embodied in (2.36) the quantum evolution here is describedby a local point to point development. Using the result

(2.37) δ(x− q(a, t)) = J−1∣∣a(x,t)

δ(a− a0(x, t)); x− q(a0, t) = 0

and evaluating the integrals (2.34) and (2.22) are equivalent to

(2.38) ρ(x, t) = J−1∣∣a(x,t)

ρ0(a(x, t)); ρ(x(a, t), t) = J−1(a, t)ρ0(a)

vi(x, t) =∂qi(a, t)

∂t

∣∣∣∣a(x,t)

; vi(x, t)|a(x,t) =∂qi(a, t)

∂t

These restate the conservation equation (2.16) and give the relations between thevelocities in the two pictures; J−1 could be called a local propagator. Now from(2.38) one can relate the accelerations in the two pictures via

(2.39)∂vi

∂t+ vj

∂vi

∂xj=

∂2qi(a, t)∂t2

∣∣∣∣a(x,t)

;(

∂vi

∂t+ vj

∂vi

∂xj

)∣∣∣∣a(x,t)

=∂2qi(a, t)

∂t2

One can now translate the Lagrangian flow equations into Lagrangian language.Differentiating (2.34) in t and using (2.35) one finds the continuity equation

(2.40)∂ρ

∂t+

∂

∂xi(ρvi) = 0


Next differentiating (2.35) and using (2.31) and (2.40) one obtains the quantumanalogue of Euler’s equation

(2.41)∂vi

∂t+ vj

∂vi

∂xj= − 1

m

∂

∂xi(V + VQ)

Finally the quasi potential condition (♠) becomes () vi = (1/m)(∂S(x, t)/∂xi).(2.38) gives the general solutions of the continuity equation (2.40) and Euler’sequation (2.41) in terms of the paths and initial density. To establish the connec-tion between the Eulerian equations and the SE note that (2.41) and () can bewritten

(2.42)∂

∂xi

(∂S

∂t+

12m

∂S

∂xi

∂S

∂xi+ V + VQ

)= 0

The quantity in brackets is thus a function of time and since this does not affectthe velocity field one may absorb it in S (i.e. set it equal to zero) leading to

(2.43)∂S

∂t+

12m

∂S

∂xi

∂S

∂xi+ V + VQ = 0

Combining all this the function ψ =√

ρexp(iS/) satisfies the SE

(2.44) i∂ψ

∂t= − 2

2m

∂2ψ

∂xi∂xi+ V ψ

This has all been deduced from (2.22) subject to the quasi potential requirement.The quantization condition (•) becomes (••)

∫C0

(∂S(x, t)/∂xi)dxi = n (n ∈ Zwhere C0 is a closed curve fixed in space that does not pass through nodes. Giventhe initial wave function ψ0(a) one can now compute ψ for all (x, t) as follows. firstsolve (2.22) subject to initial conditions q0(a) = a, ∂qi0(a)/∂t = m−1∂S0(a)/∂ato get the set of trajectories for all (x, t). Then substitute q(a, t) and qt in (2.36)to find ρ and ∂S/∂x which gives S up to an additive function of time f(t). To fixthis function up t a constant use (2.43) and one gets finally

(2.45) ψ(x, t) =√

(J−1ρ0|a(x,t)exp

[i

(∫m(∂qi(a, t)/∂t)|a(x,t)dxi + f(t)

)]The Eulerian equations (2.40) and (2.41) form a closed system of four coupledPDE to determine the four independent fields (ρ(x), vi(x)) and do not refer to thematerial paths. One notes that the Lagrangian theory from which the Euleriansystem was derived comprises seven independent fields (ρ, q(a), p(a)). In the caseof quasi potential flow there are respectively 2 or 5 independent fields. This maybe regarded as an incompleteness in the Eulerian description or a redundancy inthe Lagrangian description; it could also be viewed in terms of refinement. Onenotes also that the law of motion (2.29) for the fluid elements coincides with thatof the dBB interpretation of QM and one must be careful to discriminate betweenthe two points of view.


2.2. ELECTROMAGNETISM AND THE SE. We go next to the sec-ond paper in [474] which connects the electromagnetic (EM) fields to hydrody-namics and relates this to the quantum potential. Thus the source free Maxwellequations in free space are

(2.46) εijk∂jEk = −∂Bi

∂t; εijk∂jBk =

1c2

∂Ei

∂t; ∂iEi = ∂iBi = 0

One regards the last two equations as constraints rather than dynamical equations.First one goes to a representation of these equations in Schrodinger form and beginswith the Riemann-Silberstein 3-vector Fi =

√ε0/2(Ei + icBi) and 3 × 3 angular

momentum matrices si so that

(2.47) (si)jk = −iεijk; [si, sj ] = iεijksk

so that (2.46) is equivalent to

(2.48) i∂Fi

∂t= −ic(sj)ik∂jFk; ∂iFi = 0

To formulate groundwork for continuous representation of the spin freedoms onetransforms to a representation of the si where the z-component is diagonal via theunitary matrix

(2.49) Uai =1√2

⎛⎝ −1 i 00 0

√2

1 i 0

⎞⎠and Maxwell’s equations become

(2.50) i∂Ga

∂t= −ic(Jj)ab∂jGb; Ga = UaiFi; Ji = UsiU

−1; (a, b = 1, 0,−1)

Here one has

(2.51)

⎛⎝ G1

G0

G−1

⎞⎠ =1√2

⎛⎝ −F1 + iF2√2F3

F1 + iF2

⎞⎠ ; J1 =√2

⎛⎝ 0 1 01 0 10 1 0

⎞⎠ ;

J2 =√2

⎛⎝ 0 −i 0i 0 −i0 i 0

⎞⎠ ; J3 =

⎛⎝ 1 0 00 0 00 0 −1

⎞⎠Next one passes to an angular coordinate representation using the Euler angles(αr) = (α, β, γ) and conventions of [471] so that

(2.52) M1 = i(Cos(β)∂α − Sin(β)Ctn(α)∂β + Sin(β)Csc(α)∂γ);

M2 = i(−Sin(β)∂α − Cos(β)Ctn(α)∂β + Cos(β)Csc(α)∂γ); M3 = i∂β

The SE (2.50) becomes then

(2.53) i∂ψ(x, α)

∂t= −icMi∂iψ(x, α) ≡ i

∂ψ

∂t= −cλi∂iψ

(λi =

Mi

−i

)


where ψ is a function on the 6-dimensional manifold M = R3 ⊗ SO(3) whosepoints are labeled by (x, α). The wave function may be expanded in terms of anorthornormal set of spin 1 basis functions ua(α) (cf. [471]) in the form

(2.54) ψ(x, α, t) = Ga(x, t)ua(α) (a = 1, 0,−1); u1(α) =√

34π

Sin(α)e−iβ ;

u0(α) =ii√

32√

2πCos(α); u−1(α) =

√3

4πSin(α)eiβ ;

where∫

u∗a(α)ub(α)dΩ = δab with dΩ = Sin(α)dαdβdγ and α ∈ [0, π], β ∈

[0, 2π], γ ∈ [0, 2π]. One can show that∫

u∗aMiub(α)dΩ = (Ji)ab and multiply-

ing (2.53) (with use of (2.54)) one recovers the Maxwell equations in the form(2.50). In this formulation the field equations (2.53) come out as second orderPDE and summation over i or a is replaced by integration in αr. For example theenergy density and Poynting vector have the alternate expressions

(2.55)ε02

(E2 + c2B2) = F ∗i Fi = G∗

aGa =∫|ψ(x, α)|2dΩ;

ε0c2(E×B)i =

c

F ∗

j (si)jkFk =c

G∗

a(Ji)abGb =c

∫ψ∗(x, α)Miψ(x, α)dΩ

For the hydrodynamic model one writes ψ =√

ρexp(iS/) and splitting (2.53)into real and imaginary parts we get

(2.56)∂S

∂t+

c

λiS∂iS+Q = 0;

∂ρ

∂t+

c

∂i(ρλiS)+

c

λ(ρ∂iS) = 0; Q = −c

λi∂i√

ρ√

ρ

These equations are equivalent to the Maxwell equations provided ρ and S obeycertain conditions; in particular single valuedness of the wave function requires

(2.57)∮

C0

∂iSdxi + ∂rSdαr = n (n ∈ Z)

where C0 is a closed curve in M. In the hydrodynamic model n is interpreted as thenet strength of the vortices contained in C0 (these occur in nodal regions (ψ = 0)where S is singular). Comparing (2.56) with the Eulerian continuity equationcorresponding to a fluid of density ρ with translational and rotational freedom oneexpects

(2.58)∂ρ

∂t+ ∂i(ρvi) + λi(ρωi) = 0; vi ∼ (c/)λS; ωi ∼ (c/)∂iS

Thus one obtains a kind of potential flow (strictly quasi-potential in view of (2.57))with potential (c/)S. The quantity Q in (2.56) is of course the analogue forthe Maxwell equations of the quantum potential and will have the classical form−∇2√ρ/

√ρ when the appropriate metric on M is identified. From the Bernoulli-

like (or HJ-like) equation in (2.56) we obtain the analogue of Euler’s force law forthe EM field. Thus applying first ∂i and using (2.58) one gets

(2.59)(

∂

∂t+ vj∂j + ωj λj

)ωi = − c

∂iQ


Acting on this with λi and using [λi, λj ] = −εijkλk yields

(2.60)(

∂

∂t+ vj∂j + ωiλj

)vi = εijkωjvk −

c

λiQ

which contains a Coriolis type force in addition to the quantum contribution. Thepaths x = x(x0, α0, t) and α = α(x0, α0, t) of the fluid particles in M are obtainedby solving the differential equations

(2.61) vi(x, α, t) =∂xi

∂t; vr(x, α, t) =

∂αr

∂t

These paths are an analogue in the full wave theory of a ray.

Now one generalizes this to coordinates xµ in an N-dimensional Riemannianmanifold M with (static) metric gµν(x). The history of the fluid is encoded in thepositions ξ(ξ0, t) of distinct fluid elements and one assumes a single valued and dif-ferentiable map between coordinates (cf. Section 7.2.1). Let P0(ξ0) be the initialdensity of some continuously distributed quantity in M (mass in ordinary hydro-dynamics, energy here) and set g = det(gµν). Then the quantity in an elementaryvolume dNξ0 attached to the point ξ0 is P0(ξ0)

√−g(ξ0)dNξ0 and conservation of

this quantity is expressed via

(2.62) P (ξ(ξ0, t))√−g(ξ(ξ0, t))dNξ(ξ0, t) = P0(ξ0)

√−g(ξ0d

Nξ0 ≡

≡ P (ξ0, t) = D−1(ξ0, t)P0(ξ0); D(ξ0, t) =√

g(ξ)/g(ξ0)J(ξ0, t)

where J is the Jacobian

(2.63) J =1

N !εµ1···µn

εν1···νn∂ξµ1

∂ξν10

· · · ∂ξµN

∂ξνN0

One assumes the Lagrangian for the set of fluid particles has a kinetic term andan internal potential representing a certain kind of particle interaciton

(2.64) L =∫

P0(ξ0)(

12gµν(ξ)

∂ξµ

∂t

∂ξν

∂t− gµν c22

81

P 2

∂P

∂ξµ

∂P

∂ξν

)√−g(ξ0)dNξ0

Here is a constant with the dimension of length (introduced for dimensionalreasons) and ξ = ξ(ξ0, t); one substitutes now for P from (2.62) and writes

(2.65)∂

∂ξµ= J−1Jν

µ

∂

∂ξν0

; Jνµ =

∂J

∂(∂ξµ/∂ξν0 )

;∂ξµ

∂ξν0

Jσµ = Jδσ

ν

One assumes suitable behavior at infinity so that surface terms in the variationalcalculations vanish and the Euler-Lagrange equations are then(2.66)

∂2ξµ

∂t2+

µν σ

∂ξν

∂t

∂ξσ

∂t= −c

gµν ∂Q

∂ξν; Q =

−c

2√−gP

∂

∂ξµ

(√−ggµν ∂

√P

∂ξν

)

µν σ

=

12gµρ

(∂gσρ

∂xiν+

∂gνρ

∂ξσ− ∂gνσ

∂ξρ

)


Now one restricts to quasi-potential flows with conditions

(2.67) gµν(ξ0)∂ξµ

0

∂t=

c

∂S0(ξ0)∂ξν

0

;∮

C

∂S0(ξ0)∂ξµ

0

dξµ0 = h (n ∈ Z)

One follows the same procedures as in Section 7.2.1 so multiplying (2.66) bygσµ(∂ξσ/∂ξρ

0) and integrating gives(2.68)

gσµ(ξ(ξ0, t))∂ξσ

∂ξρ0

∂ξµ

∂t= gρµ(ξ0)

∂ξµ0

∂t+

∂

∂ξρ0

∫ t

0

(12gµν(ξ(ξ0, t))

∂ξµ

∂t

∂ξν

∂t− c

Q

)dt

Then substituting (2.67) one has

(2.69) gσµ∂ξσ

∂ξρ0

∂ξµ

∂t=

c

∂S

∂ξρ0

; S = S0 +∫ t

0

(

2cgµν

∂ξµ

∂t

∂ξν

∂t−Q

)dt

The left side gives the velocity at time t relative to ξ0 and this is a gradient. Toobtain the ξ components multiply by J−1Jρ

ν and use (2.65) to get gµν(∂ξν/∂t) =(c/)(∂S/∂ξµ) where S = S(ξ0(ξ, t), t). Thus for all time the covariant velocityof each particle is the gradient of a potential with respect to the current position.Finally to see that the motion is quasi-potential since (2.66) holds and the valuein (2.67) of the circulation is preserved following the flow, i.e.

(2.70)∂

∂t

∮C

(t)gµν∂ξν

∂tdξµ = 0

Finally for the SE one defines a fundamental link between the particle (La-grangian) and wave-mechanical (Eulerian) pictures via

(2.71) P (x, t)√−g(x) =

∫δ(x− ξ(ξ0, g))P0(ξ0)

√−g(ξ0)dNξ0

The corresponding formla for the Eulerian velocity is contained in the currentexpression

(2.72) P (x, t)√−g(x)vµ(x, t) =

∫∂ξµ

∂tδ(x− ξ(ξ0, t))P0(ξ0)

√−g(ξ0)dNξ0

These are equivalent to the following local expressions (ξ0 ∼ ξ0(x, t))(2.73)

P (x, t)√−g(x) = J−1

∣∣ξ0

P0(ξ0(x, t))√−g(ξ0(x, t)); vµ(x, t) =

∂ξµ(ξ0, t)∂t

∣∣∣∣ξ0

We can now translate the Lagrangian flow equations into Eulerian language. Firstdifferentiate (2.71) in t and use (2.72) to get

(2.74)∂P

∂t+

1√−g

∂

∂xµ(P√−gvµ) = 0

Then, differentiating (2.72) and using (2.66) and (2.74) one obtains the analogueof the classical Euler equation

(2.75)∂vµ

∂t+ vν ∂vµ

∂xν+

µν σ

vνvσ = −c

gµν ∂Q

∂xν


where Q is given by (2.66) with ξ replaced by x. Finally the quasi-potentialcondition becomes

(2.76) vµ =c

gµν ∂S(x, t)

∂xν

Formula (2.73) give the general solution of the coupled continuity and Euler equa-tions (2.74) and (2.75) in terms of the paths and initial density. To establish theconnection between the Eulerian equations and the SE note that (2.75) and (2.76)can be written

(2.77)∂

∂xµ

(∂S

∂t+

c

2gνσ ∂S

∂xν

∂S

∂xσ+ Q

)= 0

Again the quantity in brackets is a function of time which is incorporated into Sif necessary and one arrives at

(2.78)∂S

∂t+

c

2gνσ ∂S

∂xν

∂S

∂xσ+ Q = 0

Combining (2.78) with (2.74) and using (2.76) one finds for ψ =√

Pexp(i/) theequation

(2.79) i∂ψ

∂t= − −c

2√−g

∂

∂xµ

(√−ggµν ∂ψ

∂xν

)(for a system of mass /c). The quantization condition (2.70) becomes

(2.80)∮

C0

∂S(x, t)∂xµ

dxµ = n (n ∈ Z)

Now an alternate representation of the internal angular motion can be givenin terms of the velocity fields vr(x, α, t) conjugate to the Euler angles. One has

(2.81) ωi = (A−1)irvr; vr = Ariωi;

(A−1)ir =

⎛⎝ −Cos(β) 0 −Sin(α)Sin(β)Sin(β) 0 −Sin(α)Cos(β)

0 −1 −Cos(α)

⎞⎠ ;

The relations (2.52) may be written as λi = Air∂r and hence ωj λj = vr∂r. Interms of the conjugate velocities Euler’s equations (2.59) and (2.60) become

(2.82)(

∂

∂t+ vj∂j + vr∂r

)vs + Asi∂r(A−1)iqvqvr = − c

Asi∂iQ;(

∂

∂t+ vj∂j + vr∂r

)vi + εijk(A−1)krvjvr = − c

λiQ

Now one specializes the general treatment to the manifold M = R3 × SO(3) withcoordinates xµ = (xi, αr) and metric

(2.83) gµν =(

0 −1Air

−1Ari 0

); gµν =

(0 (A−1)ir

(A−1)ri 0

)and density P = ρ/3. From ∂r(

√−ggir) = 0 and gir∂r = λi one gets via[λi, λj ] = −εijkλk the relation gir(∂sgrj − ∂rgsj) = −1εijkgsk. Then the relation(2.76) becomes (2.58), (2.74) becomes (2.56), (2.75) becomes (2.82), (2.66) (with ξ


replaced by x) becomes (2.56), (2.78) becomes (2.56), (2.79) becomes (2.53), and(2.80) becomes (2.57). Writing ξµ = (qi, θr) for the Lagrangian coordinates (2.64)becomes

(2.84) L =∫

ρ0(q0, θ0)(

(A−1)ir∂qi

∂t

∂θr

∂t−Air

c2

4ρ2

∂ρ

∂qi

∂ρ

∂θr

)Sin(θ0)d3θ0d

3q0

Newton’s law (2.66) reduces to the coupled relations

(2.85)∂2qi

∂t2+ εijk(A−1)ir

∂qj

∂t

∂θr

∂t= − c

Air

∂Q

∂θr;

∂2θs

∂g2+ Asi

∂

∂θr(A−1)iq

∂θq

∂t

∂θr

∂t= − c

Asi

∂Q

∂qi

where Air is given via (2.81) with αr replaced by θr(q0, θ0, t) and one substitutesρ(q0, θ0, t) = D−1(q0, θ0, t)ρ0(q0, θ0) along with

(2.86) Q = −cAir1√

ρ

∂2√ρ

∂θr∂qi;

∂

∂qi= J−1

(Jij

∂

∂q0j+ Jis

∂

∂θ0s

);

∂

∂θr= J−1

(Jrj

∂

∂q0j+ Jrs

∂

∂θ0s

)Given the initial wavefunction ψ0(x, α) = G0a(x)ua(α) =

√ρ0exp(iS0/) one

can solve (2.85) subject to initial conditions ∂q0i/∂t = (c/)Air(∂S0/∂θ0r) and∂θ0r/∂t = (c/)Ari(θ0)(∂S0/∂q0i) to get the set of trajectories for all (q0, θ0, t).Then invert these functions and substitute q0(x, α, t) and θ0(x, α, t) in the rightside of (2.85) to find ρ(x, α, t) and in the right sides of the equations

(2.87) ∂rS =

c(A−1)ir

∂qi

∂t; ∂iS =

c(A−1)ri

∂θr

∂t

to get S up to an additive function of time f(t), which is adjusted as before (cf.(2.56)). There results(2.88)

ψ =√

D−1ρ0)|q0,θ0e[(i/c)

∫(A−1)ir(∂qi/∂t)|q0,θ0dαr+(A−1)ri(∂θr/∂t)|q0,θ0dxi+if(t)]

Finally the components of the time dependent EM field may be read off from(2.50) where

(2.89) Ga =∫

ψ(x, α)u∗a(α)dΩ

EXAMPLE 2.1. One computes the time dependence of the EM field whoseinitial form is E0i = (ECos(kz), 0, 0) with B0i = (0, (1/c)ECos(kz), 0). The initialwavefunction is ψ0 = G01u1 or

(2.90) ψ0(q0, θ0) = −√

32√

2πECos(kq03)Sin(θ01)e−iθ02

One looks for solutions to (2.85) that generate a time dependent wavefunctionwhose spatial dependence is on z alone. The Hamiltonian in the SE (2.53) then re-duces to −icM3∂3ψ(x, α) alone which preserves the spin dependence of ψ0. Hence,

3. SOME SPECULATIONS ON THE AETHER 307

since ρ is independent of θ3, the quantum potential Q in (2.86) vanishes. Somecalculation leads to

(2.91) ψ(x, α, t) = −√

32√

2πECos(k(z − ct))Sin(α)e−iβ ;

Ei = (Ecos(k(ct− z)), 0, 0); Bi = (0, (1/c)ECos(k(ct− z)), 0)Note that one obtains oscillatory behavior of the Eulerian variables from a modelin which the individual fluid elements do not oscillate! This circumvents one ofthe classical problems where it was considered necessary for the elements of acontinuum to vibrate in order to support a wave motion. Another interestingfeature is that the speed of each element |v| = |cCosc(θ01)| obeys c ≤ |v| <∞. One might regard the occurence of superluminal speeds as evidence that theLagrangian model is only a mathematical tool. Indeed performing a weighed sumof the velocity over the angles to get the Poynting vector ε0c

2(E×B)i =∫

ρvidΩthe collective x and y motions cancel to give the conventional geometrical opticsrays propagating at speed c in the z-direction.

3. SOME SPECULATIONS ON THE AETHER

We give first some themes and subsequently some details and speculations.(1) In a hauntingly appealing paper [494] P. Isaev makes conjectures, with

supporting arguments, which arrive at a definition of the aether as a Bose-Einstein condensate of neutrino-antineutrino pairs of Cooper type (Bose-Einstein condensates of various types have been considered by others inthis context - cf. [262, 263, 338, 398, 482, 510, 606, 893, 960] andRemark 5.3.1). The equation for the ψ-aether is then a solution of themassless Klein-Gordon (KG) equation (photon equation)(

2∆− 2

c2∂2

t

)ψ = 0

(cf. also [911]). This ψ field heuristically acts as a carrier of waves(playground for waves) and one might say that special relativity (SR)is a way of including the influence of the aether on physical processesand consequently SR does not see the aether (cf. here also the idea of aDirac aether in [215, 216, 302, 537, 727] and Einstein-aether theoriesas in [338, 510] - some of this is developed below). In the electromag-netic (EM) theory one looks at ψ = (φ, A) with ψi = 0 as the definingequation for a real ψ-aether, in terms of the potentials φ and A whichtherefore define the ψ-aether. EM waves are then considered as oscilla-tions of the ψ aether and wave processes in the aether accompanying amoving particle determine wave properties of the particle. The ensemblepoint of view can be considered artificial, in accord wth a conclusion wemade in [194, 197, 203], based on Theme 3 below, that uncertaintyand the ensemble “cloud” are based on the lack of deterimation of par-ticle trajectories when using the SE instead of a third order equation.Thus it is the SE context which automatically (and gratuitously) intro-duces probability; nevertheless, given the limitations of measurement, itproduces an amazingly accurate theory.

(3.1)


(2) Next there is the classical deBroglie-Bohm (dBB) theory (cf. [191, 186,187, 188, 189, 205, 203, 471] - and references in these papers) where,working from a Schrodinger equation (SE)

− 2

2m∆ψ + V ψ = iψt; ψ = ReiS/

one arrives at a quantum potential Q = −(2/2m)(∆√

ρ/√

ρ) (R =√

ρ)associated to a quantum Hamilton-Jacobi equation (QHJE)

St +(∇S)2

2m+ V + Q = 0

The ensuing particle theory exhibits trajectory motion choreographed byψ via Q = −(2/2m)(∆|ψ|/|ψ|) or directly via the guidance equation

x = v = ψ∗∇ψ

ψ∗ψ

(cf. [186, 187, 188, 203] for extensive references). Relativistic andgeometrical aspects are also provided below.

(3) In [346] Faraggi and Matone develop a theory of x−ψ duality, related toSeiberg-Witten theory in the string arena, which was expanded in vari-ous ways in [2, 41, 110, 198, 194, 191, 249, 640, 751, 958]. Here oneworks from a stationary SE [−(2/2m)∆+V (x)]ψ = Eψ, and, assumingfor convenience one space dimension, the space variable x is determinedby the wave function ψ from a prepotential F via Legendre transforma-tions. The theory suggests that x plays the role of a macroscopic variablefor a statistical system with a scaling term . Thus define a prepotentialFE(ψ) = F(ψ) such that the dual variable ψD = ∂F/∂ψ is a (linearly in-dependent) solution of the same SE. Take V and E real so that ψ = ψD

qualifies and write ∂xF = ψD∂xψ = (1/2)[∂x(ψψD)+W )] where W is theWronskian. This leads to (ψD = ψ) the relation F = (1/2)ψψ + (W/2)x(setting the integration constant to zero). Consequently, scaling W to−2i

√2m/ one obtains

i√

2m

x =

12ψ

∂F

∂ψ− F ≡ i

√2m

x = ψ2 ∂F

∂ψ2− F

which exhibits x as a Legendre transform of F with respect to ψ2. Dualityof the Legendre transform then gives also

F = φ∂φ

(i√

2m x

)−(

i√

2m x

); φ = ∂ψ2F =

ψ

2ψ

so that F and (i√

2m x/) form a Legendre pair. In particular one has ρ =|ψ|2 = 2i

√2m

x+2F which also relates x and the probability density (butindirectly since the x term really only cancels the imaginary part of 2F).In any event one sees that the wave function ψ specifically determines theexact location of the “particle” whose quantum evolution is described by

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)


ψ. We mention here also that the (stationary) SE can be replaced by athird order equation

4F′′′ + (V (x)− E)(F′ − ψF′′)3 = 0; F

′ ∼ ∂F

∂ψ

and a dual stationary SE has the form

2

2m

∂2x

∂ψ2= ψ[E − V ]

(∂x

∂ψ

)3

A noncommutative version of this is developed in the second paper of[958].

(4) We also note for comparison and analogy some relations between Le-gendre duals in mechanics, thermodynamics, and (x, ψ) duality. Thus(cf. [202, 596]) one has in mechanics px−L = H via L = (/2)mx2 − Vand H = (p2/2m) + V with p = ∂L/∂x and x = ∂H/∂p. In thermody-namics one has a Helmholtz free energy F with F = U−TS for energy U,entropy S, and temperature T. Set F = −F to obtain F = T (∂F/∂T )−Uand U = S∂SU − F (where ∂TF = S and ∂SU = T ). Now put this in atable where we write the (x, ψ) duality in the form χ = ψ2(∂F/∂ψ2)− F

with F = φ(∂χ/∂φ) − χ (for χ = (i√

2m/)x and φ = (∂F/∂ψ2)). Thisleads to a table

Mechanics Thermodynamics (x, ψ) dualityx, p, L, H T, S, F , U ψ2, φ, F, χ

px−H = L TS − U = F ψ2φ− F = χ

L = x∂H∂p −H F = S ∂U

∂S − U F = φ∂χ∂φ − χ

H = p∂L∂x − L U = T ∂F

∂T −F χ = ψ2 ∂F∂ψ2 − F

One says that e.g. (F, χ) or (F , U) or (L,H) form a Legendre dual pair and in thefirst situation one refers to (x, ψ) duality. One sees in particular that F = ψ2

−χ where ψ2φ ∼ xp in mechanics. Note that φ = ∂F/∂ψ2 = (1/2ψ)(∂F/∂ψ) =ψ/2ψ with ψ2φ = (1/2)ψψ = (1/2)|ψ|2. In any event χ = −i(

√2m/)x and we

will see below how the physics can be expressed via ψ, ∂/∂ψ, dψ etc. withoutmentioning x. This allows one to think of the coordinate x as an emergent entityand we like to think of x− ψ duality in this spirit.

3.1. DISCUSSION OF A PUTATIVE PSI AETHER. We mention[650, 753, 935] for some material on the aether and the vacuum and refer tothe bibliography for other references. We sketch first some material from [2, 41,110, 249, 751, 958] which extends theme 3 to the Klein-Gordon (KG) equation.Following [958] take a spacetime manifold M with a metric field g and a scalarfield φ satisfying the KG equation. Locally one has cartesian coordinates xα (α =0, 1. · · · , n− 1) in which the metric is diagonal with gαβ(x) = ηαβ(x) and the KG

equation has the form (x+m2)φ(x) = 0 (x?∼ (2/c2)[(∂2

t /c2)−∇2] - cf. (2.26)).Defining prepotentials such that φ(α) = ∂F(α)[φ(α)]/∂φ(α) where φ(α) and φ(α) aretwo linearly independent solutions of the KG equation depending on a variable xα

(where the xβ for β = α enter φα and φα as parameters) one has as above (with

(3.7)

(3.8)

(3.9)

φ


a different scaling factor)√

2m

xα =

12φ(α) ∂F(α)[φα)]

∂φ(α)

This is suggested in [346] and used in [958]; the factor√

2m/ is simply a scalingfactor and it may be more appropriate to scale x0 ∼ ct differently or in fact to scaleall variables as indicated below with factors βi(xj , t). Locally now F(α) satisfiesthe third order equation

4F(α)′′′

+ [V (α)(xα) + m2](φ(α)F

(α)′′− F

(α)′))3 = 0

where ′ ∼ ∂/∂φ(α) and the “effective” potential V has the form

V (α)(xα) =

⎡⎣ 1φ(x)

n−1∑β=0, β =α

∂β∂βφ(x)

⎤⎦∣∣∣∣∣∣xβ =α fixed

REMARK 7.3.1. Strictly speaking V α does not have the form R/R of aquantum potential; however since it is created by the wave function φ we couldwell think of it as a form of quantum potential. We will refer to it as the effectivepotential as in [346] and note from Section 3.2 that with ηµν = (−1, 1, 1, 1) and = −(1/c2)∂2

t + ∆, one has for φ = Rexp(iS/)

12m

(∂S)2 =2

2m

φ

φ+

2

2m

R

R;

∂(R2∂S) = 0; Qrel = − 2

2m

R

RThe discussion below indicates that much further development of these themesshould be possible.

As indicated in [346], once xα is replaced with its functional dependence onFα α(φα);further the functional structure of Fα does not depend on the parameters xβ forβ = α (which enter φα

∂

∂xα=

(8m)1/2

1E(α)

∂

∂φ(α); dxα =

(8m)1/2E(α)dφ(α)

where E(α) = φ(α)F(α)′′−F(α)′

on the spaces TP (U) and T ∗P (U) (P ∈ U - local tangent and cotangent spaces).

Now using the linearity of themetric tensor field (cf. [322]) one sees that the components of the metric in the(φ(α),F(α) are

Gαβ(φ) =2

8mE(α)E(β)ηαβ(x)

Now let zµ (µ = 0, 1, · · · , n − 1) be a general coordinate system in U and writethe coordinate transformation matrices via

Aαµ =

∂xα

∂zµ; (A−1)µ

α =∂zµ

∂xα

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(α) α∂α−V α φ

α)∂− F ; ( = 0

Note there is no summation over α in (3.14).

given in (3.10), (3.11) becomes an ordinary differential equation for F

as parameters). Now as a consequence of (3.10) one has

. Here (3.14) represents an induced parametrization


The metric then takes the form

gµν(z) =8m

2

1E(α)E(β)

AαµAβ

νGαβ(φ)

The components of the metric connection can be computed via

Γρµν =

12gρσ(z)

∑P

εPP [∂gσν(z)/∂zµ]

where P is a cyclic permutation of the ordered set of indices σνµ and εP is the(α) depends

on all the zµ. (α),F(α)parametrization via

Γρµν =

(2m

)1/2E(ρ)E(σ)

E(γ)(A−1)ρ

τ (A−1)σχGτχ×

×∑P

εPP[Aγ

µ

∂

∂φ(γ)

(1

E(α)E(β)Aα

σAβνGαβ

)]In [958] one computes, in the (φ(α),F(α)) parametrization, the components of thecurvature tensor, the Ricci tensor, and the scalar curvature and gives an expressionfor the Einstein equations (we omit the details here). These matters are taken upagain in [41] for a general curved spacetime and some sufficient constraints areisolated which make the theory work. Also in both papers a quantized version ofthe KG equation is also treated and the relevant x − ψ duality is spelled out inoperator form. We omit this also in remarking that the main feature here for ourpurposes is the fact that one can describe spacetime geometry (at least locally) interms of (field) solutions of a KG equation and prepotentials (which are themselvesfunctions of the fields). In other words the coordinates are programmed by fieldsand if the motion of some particle of mass m is involved then its coordinatesare choreographed by the fields with a quantum potential entering the picture

coordinates and to connect this with the aether idea one should examine the aboveformulas for m→ 0.

Thus (cf. [191, 198, 206, 346]) one writes (in 1-D)

∂ψF = ψD ∼ ψ; ∂xF = ∂ψF∂xψ = ψDψx =12[∂x(ψDψ) + W ]; W = ψDψx − ψD

x ψ

and W = constant (this is the scaling factor). For example with x ∼ ct we write

F =12ψEψ + Wct =

12ψDψ + γct

to find (χ0 ∼ γct)

γct =12φ0 ∂F0

∂φ0− F

0; E0 = φ0 ∂2F0

∂(φ0)2− ∂F0

∂φ0

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

signature of P. Via the coordinate transformation (3.16) the function φThe metric connection (3.18) can be expressed in the φ

via (3.12). In [2] a similar duality is worked out for the Dirac field and cartesian

Let us do some rescaling now and recall the origin of equations such as (3.10).


dt =E0dφ0

2γc; ∂t =

2γc

E0

∂

∂φ0

For the other variables x1, x2, x3 we write γc = β and

dxi =Eidφi

2β;

∂

∂xi=

2β

Ei

∂

∂φi

√2m/ by γc when ct ∼

t = x0 and by βi for the xi (1 ≤ i ≤ 3) to obtain for example (here βi = βi(xj , t)and β0 = γc)

Gασ(φ) =4EαEσ

βαβσηασ

in place of (2.15). Consequently we obtain an heuristic result

THEOREM 3.1. One can then continue this process to find analogues of2t /c2)−∇2

x+(c2m2/2)]φ =0 so letting m → 0 we obtain the photon (or aether) equation (2.10) of Isaev (as-suming the scaling factors βi can be taken independently of m). Now however we

Let us sketch next some arguments from [494] where we omit the historical andphilosophical introduction describing some opinions and ideas of famous people,e.g. Dirac, Einstein, Faraday, Lorentz, Maxwell, Planck, Poincare, Schwinger, etal. One begins with a KG equation(

2∇2 − 2

c2

∂2

∂t2−m2c2

)ψ(s, t) = 0 ≡ (2−m2c2)ψ = 0

One asserts that any relativistic equation for a free particle with mass m whouldbe understood not as an equation in vacuum but as an equation for a particlewith mass m in the aether; thence setting the mass equal to zero one arrives at

This is called the ψ-aether in contrastto the (impossible!) Lorentz-Maxwell aether. Now consider the case of an EMfield with H = curl(A) and E = −(1/c)∂tA−∇φ and use the Lorentz conditiondivA + (1/c)∂tφ = 0. Then the potentials A and φ satisfy

A = ∇2A− 1c2

∂2A∂t2

= 0; φ = ∇2φ− 1c2

∂2φ

∂t= 0

Using the Lorentz gauge one can take φ = 0 so the charge independent part of thepotentials is determined via

A = 0; divA = 0; φ = 0; E = −1c

∂A∂t

; H = curlA

the general solution is given by a superposition of transverse waves. For a more

i

0; these are called the equations for the real ψ-aether.

The classical unphysicality of φ, A now is removed in attaching them to the

(3.23)

(3.24)

(3.25)

(3.16) - (3.19) and we think now of the KG equation as [(∂

have in addition a geometry for this putative aether via (3.22) - (3.25) and their

(3.26)

(3.1) for the equation of the aether itself.

(3.27)

(3.28)

This system (3.28) is completely equivalent to the Maxwell-Lorentz equations and

symmetric representation one can write ψ = (φ,A) and (3.27) becomes ψ (x, t) =

continuations (see [1021]).

Again (3.11) holds along with (3.12). Now simply replace

physically observable reality of the ψ-aether. Indeed the KG equation can bewritten as a product of two commuting matrix operators

Iαβ(−m2) =∑

δ

(iγn ∂

∂xn+ m

)αδ

(iγk ∂

∂xk−m

)δβ

and in order that the field function satisfy the KG equation one could require thatit satisfy also one of the first order equations(

iγn ∂

∂xn+ m

)ψ = 0 or

(iγn ∂

∂xn−m

)ψ = 0

field (there may be some question about m = 0 here). Recall that particle solutionsof the KG equation corresponding to single valued representations of the Lorentzgroup have integer spins while particles with half-integer spin are described bya spinor representation. One also knows that the neutrino has spin /2. In any

but are connected to physical reality in the form of the ψ-aether by neutrino-anti-neutrino pairs (cf. [494] for further arguments along these lines).

For an interesting connection of the ψ-aether with QM consider a hydrogenatom with spherically symetric and time independent potential V (r) = V (r) wherer = |r|. The solution to the SE −i∂tψ = −(2/2m)∇2ψ + V (r)ψ is obtained byseparation of variables ψ = u(r)f(t) with u(r) = R(r)Y (θ, φ). This is a problemof two body interaction (a proton and an electron) and for stationary states withenergy E one looks at ψ(x, t) = Cexp(−iEt/) satisfying

1sin(θ)

∂

∂θ

(sin(θ)

∂Y

∂θ+

1sin2(θ)

∂2Y

∂φ2

)+ λY = 0;

1r2

d

dr

(r2 dR

dr

)+

2µ

2[E − V (r)]− λ

r2

R = 0

Here µ is the reduced mass of the system (proton + electron), E is the energy level2

is solved by further separation of variables Y = Θ(θ)Φ(φ) leading to

∂2Φ∂φ2

+ νΦ = 0;1

sin(θ)d

dθ

(sin(θ)

∂Θ∂θ

)+(λ− ν

sin2θ

)Θ = 0

The solution for Φ is Φm(φ) = (1/2π)exp(imφ) with ν = m2 and physically

with |m| ≤ . For R one has

1r2

d

dr

(r2 dR

dr

)+

2µ

2

e2

rR(r) +

2µ

2ER(r)− ( + 1)

r2R = 0

Now here V (r) = (2µ/2)(e2/r) = [( + 1)/r2] and the term involving e2/r isresponsible for the Coulomb interaction of a proton with an electron; however thesecond term (+1)/r2 does not depend on any physical interaction (even thoughin [847] it is said to be connected with angular momentum). Now putting theCoulomb interaction to zero the (+ 1)/r2 term does not disappear and it makes

(3.29)

(3.30)

Putting m = 0 in (3.30) one has possible equations for the neutrino-anti-neutrino

event (cf. (3.27)-(3.28)) the potentials φ and A are not merely auxilliary functions

(3.31)

admissible solutions for Θ (associated Legendre polynomials) require λ = ( + 1)

(3.32)

(3.33)

for the bound state p + e (E < 0), and V (r) = e /r is the potential energy. (3.31)


no sense to attribute it to angular momentum. It is now claimed that in fact thisterm arises because of the ψ-aether and an argument based on standing waves ina spherical resonator is given. Thus following [155] one considers an associatedBorgnis function U(r, θ, φ), having definite connections to E and H, and when itsatisfies

∂2U

∂r2+

1r2sin(θ)

[∂

sin(θ)∂U

∂θ+

∂

∂φ

1sin(θ)

∂U

∂φ

]+ k2U = 0

the Maxwell equations are also valid. Further U is connected by definite relationswith A and φ, i.e. with the ψ-aether (presumably all this is spelled out in [155]).

1 2

there results

(A)1

sin(θ)∂

∂θsin(θ)

∂F2

∂θ+

1sin2θ

∂2F2

∂φ2+ γF2 = 0

(B) r2 ∂2F1

∂r2+ k2r2F1 − γF1 = 0

One considers here EM waves harmonic in time and characterized either by thefrequency ν = kc/2π or by the wave vector k = 2πν/c with [k] = 1/cm. Now

Setting F1(r) = rf(r) oneobtains then

d2f

dr2+

2r

df

dr+[k2 − n(n + 1)

r2

]f(r) = 0

d2R

dr2+

2r

dR

dr+(

2µE

2+

2µe2

2r− ( + 1)

r2

)R = 0

2 2 2

d2R

dr2+

2r

dR

dr+(

k2 − ( + 1)r2

)R = 0

where 2µ2p2/2µ2 = k22/2 = k2

are identical and are solved under the same boundary conditions (i.e. f(r) shouldbe finite as r → 0 and when r → ∞ one wants f(r) → 0 on the boundary ofa sphere).the sphere at values n = 0, 1, · · · with m ≤ n. Since EM waves are nothingbut oscillations of the ψ-aether the term n(n + 1)/r2

standing waves of the ψ-aether in a sphere resonator. Thus (mathematically atleast) one can say that the problem of finding the energy levels in a hydrogenatom via the SE is equivalent to the problem of finding natural EM oscillationsin a spherical resonator. One recalls that one of the basic postulates of QM(quantization of orbits in a hydrogen atom a la Bohr with mvr = n/2π) isequivalent to determination of conditions for existence of standing waves of theψ-aether in a spherical resonator. This suggests that QM may be equivalent to“mechanics” of the ψ-aether. One remarks that until now only a small part ofthe alleged ψ-aether properties have been observed, namely in superfluidity and

(3.34)

(3.35)

(A) in (3.35) is the same as (3.31) with spherical function solutions and regular

∂θ

solutions of (B) in (3.35) exist when γ = n(n + 1).

(3.36)

(3.37)

(3.38)

with k the wave vector). Now (3.36) and (3.38)

The corresponding solutions to (3.36) represent standing waves inside

in (3.36) is responsible for

To solve (3.34) one writes U = F (r)F (θ, φ) (following the notation of [155]) and

Setting 2µe / r = 0 and replacing E by E = p /2µ in (3.37) one obtains

A little calculation puts (3.33) into the form

4. REMARKS ON TRAJECTORIES 315

superconductivity (see e.g. [968]). It is suggested that one might well rethink a lotof physics in terms of the aether, rather than, for example, the standard model.In any event there is much further discussion in [494], related to real physicalsituations, and well worth reading.

4. REMARKS ON TRAJECTORIES

There have been a number of papers written involving microstates and Bohmianmechanics (cf. [138, 140, 139, 191, 194, 197, 203, 305, 306, 307, 308, 309,347, 373, 374, 375, 376, 348, 349, 520]) and we sketch here some features ofthe Bouda-Djama method following [309]. There are some disagreements regard-ing quantum trajectories, discussed in [138, 375], which we will not deal withhere. Generally we have followed [347] in our previous discussion and microstateswere not explicitly considered (beyond mentioning the third order equation andthe comments in Remark 2.2.2). Thus, referring to [309] for philosophy, one be-gins with the SE −(2/2m)∆ψ + V ψ = iψt where ψ = Rexp(iS/) in 3-D andarrives at the standard

(4.1)1

2m(∇S)2 − 2

2m

∆R

R+ V = −St; ∇ ·

(R2∇S

m

)+ V = −∂(R2)

witeh Q = −(2/2m)(∆R/R). Then one sets

(4.2) j =

2mi(ψ∗∇ψ − ψ∇ψ∗) = R2∇S

m⇒ ∇ · j + ∂tR

2 = 0

and ρ = |ψ|2 = R2 as usual. The velocity v is taken as v = j/ρ = ∇S/m herein the spirit of Bohm (and Durr, Goldstein, Zanghi, et al). Working in 1-D withS = S0(x,E) − Et one recovers the stationary HJ equation of Section 2.2 forexample and there is some discussion about the situation S0 = constant referringto Floyd and Farragi-Matone. Explicit calculations for microstates are consideredand comparisons are indicated. The EP of Faraggi-Matone is then discussed as inSection 2.2 and the quantum mass field mQ = m(1 − ∂EQ) is introduced. Thisleads to the third order differential equation for x (where P = ∂xS0 = mQx)

(4.3)m2

Q

2m+ V (x)− E +

|hbar2

4m

(m′′

Q

mQ− 3

2(m′

Q)2

m2Q

−m′

Q

mQ

x

x2+

...x

x3− 5

2x2

x4

)= 0

It is observed correctly that (4.3) is a difficult equation to manipulate, requiringa priori a solution of the QSHJE.

Now one proposes a Lagrangian which depends on x, x and the set of hiddenvariables Γ which is connected to constants of integration from an equation like(4.3). This approach was developed in order to avoid dealing with the Jacobitype formula t − t0 = ∂S0/∂E which the authors felt should be restricted to HJequations of first order. Then one looks for a quantum Lagrangian Lq such that(d/dt)(∂Lq/∂x)− ∂xLq = 0 and writes

(4.4) Lq(x, x,Γ) =m

2x2f(x,Γ)− V (x);

∂Lq

∂x= mxf(x,Γ);

∂Lq

∂x=

m

2x2fx − Vx


This leads to

(4.5) mf(x,Γ)x +mx2

2fx + Vx = 0

Then set Hq = (∂xLq)x− Lq and P = ∂Lq/∂x = mxf so

(4.6) Hq =mx2

2f(x,Γ) + V (x) =

P 2

2mf+ V (x)

Working with the stationary situation S = S0(x,Γ) − Et some calculation givesthen

(4.7)1

2mfS2

x + V = −St ⇒1

2mf(∂xS0)2 + V (x)− E = 0

Now referring to the general equation (2.18) in Chapter 2 (extracted from [347])one writes here w = θ/φ ∼ ψD/ψ ∈ R with (α ∼ ω) so that (cf. [?, ?])

(4.8) e2iS0/ = eiω (θ/φ) + i

(gt/φ)− i S0 = Tan−1 θ + µφ

νθ + φ

(cf. [139] for details). For the QSHJE the basic equation is (2.2.17) which werepeat as

(4.9)1

2m (S′0)

2 + W + Q = 0; W = − 2

4m

e2iS0/, x

∼ V − E; Q = 2

4mS0, x

There is a “quantum” transformation x → x described in [347, 348] with theQSHJE arising then from a conformal modification of the CSHJE. Thus note(•) x, S0 = −(S′

0)−2S0, x and define U(S0) = x, S0/2 = −(1/2)(S′

0)−2S0, x.

This gives a conformal rescaling 12m (S′

0)2[1− 2U] + V − E = 0 since

(4.10)1

2m(S′

0)2[1− 2

U] =1

2m(S′

0)2[1− 2

2x, S0] =

12m

(S′0)

2[1 +2

2(S′

0)−2S0, x] =

=1

2m(S′

0)2 +

2

4mS0, x =

12m

(S′0)

2 + Q ⇒ Q = − 2

2m(S′

0)2U

which agrees with Q = (2/4m)S0, x using (•). Then from (4.10)

(4.11)(

∂x

∂x

)2

= 1 = 2U(S0) = 1 + 2m(S′

0)2Q ⇒ x =

∫ x dx√1 + 2m(∂S0)−2Q

Similarly using the QSHJE (2.2.17) we have

(4.12)(

∂x

∂x

)2

= (S′0)

−2[(S′0)

2 + 2mQ] = [(S′0)

2 − 2mW](S′0)

−2 ⇒

⇒ x =∫ x S′

0dx√2m(E − V )

4. REMARKS ON TRAJECTORIES 317

This all follows from [347, 348] and is used in [309]. Now from (4.7), (4.9), (4.10),and the QSHJE one can write (correcting a sign in [309])

(4.13) f(x,Γ) =[1 +

2

2(S′

0)−2S0, x

]−1

⇒ f =(S′

0)2

2m(E − V )

and via (4.8) Γ = Γ(E,µ, ν) with f = f(x, E, µ, ν). Putting this in (4.7) givesthen

(4.14) E =mx2

2(S′

0)2

2m(E − V )+ V ⇒ xS′

0 = 2(E − V )

Note that this equation also follows from (4.5), namely mxf = ∂Lq/∂x, andintegration (cf. [309]). Now for the appropriate third order trajectory equationin this framework, one finds from (4.14) and the QSHJE

(4.15) (E − V )4 − mx2

2(E − V )3 +

2

8

[32

(x

x

)2

−...x

x

](E − V )2−

−2

8

[x2 d2V

dx2+ x

dV

dx

](E − V )− 32

16

[x

dV

dx

]2

= 0

(cf. [138, 309]). This is somewhat simpler to solve that (4.3) since it is indepen-dent of the SE and the QSHJE. We refer now to [138, 140, 139, 305, 306, 307,308, 309] for more in this direction.

CHAPTER 8

REMARKS ON QFT AND TAU FUNCTIONS

1. INTRODUCTION AND BACKGROUND

The idea here is to connect various aspects of quantum groups, quantum fieldActually the

are already in place but we want to make matters explicitFor references we have in mind

[191, 205] for Wick’s theorem and tau functions, [157, 158, 159, 160, 161, 162,168] for quantum groups and QFT, and [147, 192, 193, 207, 212, 656, 662,664] for q-tau functions and Hirota formulas.

We go first to the beautiful set of ideas in [157, 158, 159, 161] (and ref-erences there) and will sketch some of the results (sometimes without proof).The main theme is to create an algebraic framework which “tames” the intri-cate combinatorics of QFT. Thus the main concepts used in the practical calcu-lation of observables are covered, namely, normal and time ordering and renor-malization. One begins with a finite dimensional vector space V (ei a basis)and then the symmetric algebra S(V ) is defined as S(V ) = ⊕∞

0 Sn(V ) whereS0(V ) = C, S1(V ) = V, and Sn(V ) is spanned by elements ei1 ∨ · · · ∨ ein

withi1 ≤ i2 ≤ · · · ≤ in. The symbol ∨ denotes the associative and commutativeproduct ∨ : Sm(V ) ⊗ Sn(V ) → Sm+n(V ) defined on elements of the basis via(ei1 ∨ · · · ∨ eim

) ∨ (eim+1 ∨ · · · ∨ eim+n) = eiσ(1) ∨ · · · ∨ eiσ(m+n) where σ is the

permutation on m + n elements such that iσ(1) ≤ · · · ≤ iσ(m+n) (extended bylinearity and associativity to all elements of S(V ). The unit of S(V ) is 1 ∈ C(i.e. for any u ∈ S(V ) one has 1 ∨ u = u ∨ 1 = u). The algebra is graded via|u| = n for u ∈ Sn(V ) and ∨ is a graded map (i.e. if |u| = n and |v| = m then|u∨ v| = m + n). In fact S(V ) is the algebra of polynomials in ei with elements ofSn(V ) being homogeneous polynomials of degree n. Let now u, v, w be elementsof S(V ), a, ai, b, c ∈ V and ei basis elements of V as above. Define a coproductover S(V ) via

(1.1) ∆1 = 1⊗ 1; ∆a = a⊗ 1 + 1⊗ a; ∆(u ∨ v) =∑

(u1 ∨ v1)⊗ (u2 ∨ v2)

(here e.g. ∆u =∑

u1 ⊗ u2 etc.). This coproduct is coassociative and cocommu-tative and equivalent to a coproduct from [768]. Note in particular

(1.2) ∆(a ∨ b) =∑

(a1 ∨ b1)⊗ (a2 ∨ b2)

319

theory (QFT), and tau functions by various algebraic techniques.

and direct with some attempt at understanding.connectionsprincipal

320 8. REMARKS ON QFT AND TAU FUNCTIONS

and since ∆a = a⊗ 1 + 1⊗ a with ∆b = b⊗ 1 + 1⊗ b one obtains

(1.3) ∆(a ∨ b) = (a ∨ b)⊗ 1 + 1 + a⊗ b + b⊗ a + 1⊗ (a ∨ b)

At the next order ∆(a ∨ b ∨ c) = 1⊗ a ∨ b ∨ c + a⊗ b ∨ c + b⊗ a ∨ c + c⊗ a ∨ b +a ∨ b⊗ c + a ∨ c⊗ b + b ∨ c⊗ a + a ∨ b ∨ c⊗ 1. Generally if u = a1 ∨ a2 ∨ · · · ∨ an

one has following [610]

(1.4) ∆u = u⊗ 1 + 1⊗ u +n−1∑p=1

∑σ

aσ(1) ∨ · · · ∨ aσ(p) ⊗ aσ(p+1) ∨ · · · ∨ aσ(n)

where σ runs over the (p, n − p) shuffles. Such a shuffle is a pemutation σ of1, · · · , n such that σ(1) < σ(2) < · · · < σ(p) and σ(p + 1) < · · · < σ(n). Thecounit is defined via ε(1) = 1 and ε(u) = 0 if u ∈ Sn(V ) for n > 0. The antipodeis defined via s(u) = (−1)nu for u ∈ Sn(V ) and in particular s(1) = 1. Since thesymmetric product is commutative the antipode is an algebra morphism(s(u∨v) =s(u) ∨ s(v)). This algebra is called the symmetric Hopf algebra.

One goes next to the Laplace pairing on S(V ) defined as a bilinear formV × V → C denoted (a|b) and extended to S(V ) via

(1.5) (u ∨ v|w) =∑

(u|w1)(v|w2); (u|v ∨ w) =∑

(u1|v)(u2|w)

A priori the bilinear form is neither symmetric nor antisymmetric (cf. [433] wherethe form is still more general). The term Laplace pairing comes from the Laplaceidentities in determinant theory where the determinant is expressed in terms ofminors. In [433] it is proved that if u = a1 ∨ a2 ∨ · · · ak and v = b1 ∨ b2 ∨ · · · ∨ bn

then (u|v) = 0 if n = k and (u|v) = perm(ai|bj) if n = k where

(1.6) perm(ai|bj) =∑

σ

(a1|bσ(1)) · · · (ak|bσ(k))

over all permutations σ of (1, · · · , k). The permanent is a kind of determinantwhere all signs are positive. For example (a ∨ b|c ∨ d) = (a|c)(b|d) + (a|d)(b|c).A Laplace-Hopf algebra now means the symmetric Hopf algebra with a Laplacepairing.

Now comes the circle product which treats in one stroke the operator productand the time ordered product according to the definition of the bilinear form (a|b).Following [804] the circle product is defined via v v =

∑u1 ∨ v1(u2|v2) which by

cocommutativity of the coproduct on S(V ) is equivalent to

(1.7) u v =∑

u1 ∨ v2(u2|v1) =∑

u2 ∨ v1(u1|v2) =∑

(u1|v1)u2 ∨ v2

A few examples are instructive, thus

(1.8) a b = a ∨ b + (a|b); (a ∨ b) c = a ∨ b ∨ c + (a|c)b + (b|c)a; a (b ∨ c) =

= a ∨ b ∨ c + (a|c)b + (a|b)c; a b c = a ∨ b ∨ c + (a|b)c + (a|c)b + (b|c)aFurther some useful properties are

(1.9) u 1 = 1 u = u; u (v + w) = u v + u w; u (λv) = (λu) v = λ(u v)

1. INTRODUCTION AND BACKGROUND 321

A detailed proof of the associativity of is given in [158] which is based on twolemmas which we record as (u|v w) = (u v|w) and

(1.10) ∆(u v) =∑

(u1 ∨ v1)⊗ (u2 v2) =∑

(u1 v1)⊗ (u2 ∨ v2)

The proofs are straightforward and associativity of follows directly. One hasalso a useful formula (proved in [158])

(1.11) u ∨ v =∑

(s(u1)|v1)u2 v2 =∑

(u1|s(v1))u2 v2

Next in [158] one proves a few additional formulas, namely

(1.12)(u|v) =

∑s(u1 ∨ v1) ∨ (u2 v2); u (v ∨ w) =

∑(u11 v) ∨ (u12 w) ∨ s(u2)

1.1. WICK’S THEOREM AND RENORMALIZATION. One showsnext that the circle product satisfies a generalized version of Wick’s theorem.There are two such theorems for normal or time ordered products but they have anidentical structure (cf. [935]). In verbal form one has e.g. that the time or normalordered product of a given number of elements of V is equal to the sum over allpossible pairs of contractions (resp. pairings). A contraction a•b• is the differencebetween the time ordered and the normal product. Thus a•b• = T (ab)− : ab :.A pairing ab is the difference between the operator product and the normalproduct, namely ab = ab− : ab :. Both the contraction and the pairing arescalars and one can identify the normal product : ab : with a∨b because the normalproduct has all the properties required for a symmetric product (see remarkslater). Now on the one hand the time ordered product is symmetric and a•b• isa symmetric bilinear form that we can identify with (a|b). Then the time orderedproduct of two operators is equal to the circle product obtained from the symmetricbilinear form. On the other hand the pairing obtained from the operator productis an antisymmetric bilinear form ab = (ab−ba)/2 that we can also identify with(a different) (a|b). The operator product of two elements of V is now the circleproduct obtained with this antisymmetric bilinear form.

so ab = ab− : ab :∼ ab = a b = a ∨ b + (a|b) with (a|b) = ab. Similarly fora•b• = (a|b) = T (ab)− a ∨ b ∼ T (ab) = (a|b) + a ∨ b with T (ab) = a b.

Now the main ingredient in proving Wick’s theorem as in [995] is to go froma product of n elements of V to a product of n + 1 elements via

(1.13) : a1 · · · an : b =: a1 · · · anb : +n∑

j=1

a•j b

• : a1 · · · aj−1aj+1 · · · an :

To prove this we use the definition of the circle product and (1.1) to get

(1.14) u b = u ∨ b +∑

(u1|b)u2; u = u1 ∨ · · · ∨ an

(note ∆b = b⊗1+1⊗b and ub =∑

(u1|b1)u2∨b2 =∑

(u1|b)u2∨1+∑

(u1|1)u2∨b;but w ∨ 1 = w and u 1 = u =

∑(u1|1)u2 since ∆1 = 1⊗ 1). The Laplace pairing

(u1|b) is zero if the grading of u1 is different from 1, so u1 must be an element of V.

REMARK 8.1.1. To put this in a more visible form recall a b = a∨ b + (a|b)

According to the general definition (1.4) for ∆u this happens only for the (1, n−1)shuffles. By definition then a (1, n − 1) shuffle is a permutation σ of (1, · · · , n)such that σ(2) < · · · < σ(n) and the corresponding terms of the coproduct in ∆uare

∑n1 aj ⊗ a1 ∨ · · · ∨ aj−1 ∨ aj+1 ∨ · · · ∨ an. This proves the required identity so

the circle product satisfies Wick’s theorem. One repeats that if the bilinear formis 1/2 of the commutator then one has the Wick theorem for operator productsand if the bilinear form is the Feynman propagator then one has Wick’s theoremfor time ordered products.

Now one introduces a renormalised circle product . In a finite dimensionalvector space the purpose of renormalisation is no longer to remove infinities, sinceeverything is finite, but to provide a deformation of the circle product involvingan infinite number of parameters. The physical meaning of the parameters goesas follows. Time ordering of operators is clear when two operators are defined atdifferent times; however the meaning is ambiguous when the operators are definedat the same time. Renormalisation is present to parametrise the ambiguity. Theparameters are defined as a linear map ζ : S(V ) → C such that ζ(1) = 1 andζ(a) = 0 for a ∈ V . The parameters form a renormalisation group with product ∗defined via

(1.15) (ζ ∗ ζ ′)(u) =∑

ζ(u1)ζ ′(u2)

This is called the convolution of the Hopf algebra. The coassociativity and co-commutativity of the Hopf algebra implies that ∗ is associative and commutative.The unit of the group is the counit ε of the Hopf algebra and the inverse ζ−1 isdefined via (ζ ∗ ζ−1)(u) = ε(u) or recursively via

(1.16) ζ−1(1) = 1; ζ−1(u) = −ζ(u)−′∑

ζ(u1)ζ−1(u2)

where∑′

u1 ⊗ u2 = ∆u − 1 ⊗ u − u ⊗ 1 for u ∈ Sn(V ) with n > 0. As examplesnote

(1.17) ζ−1(a) = 0; ζ−1(a ∨ b) = −ζ(a ∨ b); ζ−1(a ∨ b ∨ c) = −ζ(a ∨ b ∨ c);

ζ−1(a∨b∨c∨d) = −ζ(a∨b∨c∨d)+2ζ(a∨b)ζ(c∨d)+2ζ(a∨c)ζ(b∨d)+2ζ(a∨d)ζ(b∨c)Now from the renormalization parameters one defines a Z-pairing

(1.18) Z(u, v) =∑

ζ−1(u1)ζ−1(v1)ζ(u2 ∨ v2)

A few examples are(1.19)Z(u, v) = Z(v, u); Z(1, u) = ε(u); Z(a, b) = ζ(a ∨ b); Z(a, b ∨ c) = ζ(a ∨ b ∨ c);

Z(a ∨ b, c ∨ d) = ζ(a ∨ b ∨ c ∨ d)− ζ(a ∨ b)ζ(c ∨ d)The main property of the Z-pairing is the coupling identity

(1.20)∑

Z(u1 ∨ v1, w)Z(u2, v2) =∑

Z(u, v1 ∨ w1)Z(v2, w2)

One can check e.g. when u = a, v = b, w = c ∨ d that

(1.21) Z(a ∨ b, c ∨ d) = Z(a, b ∨ c ∨ d) + Z(a, c)Z(b, d) + Z(b, c)Z(a, d)


To show (1.20) one uses the lemma∑

ζ−1(u1 ∨ v1)Z(u2, v2) = ζ−1(u)ζ−1(v) (cf.[158] for proof). To prove the coupling identity one expands Z via(1.22)∑

Z(u1 ∨ v1, w)Z(u1, v2) =∑

ζ−1(u11 ∨ v11)ζ−1(w1)ζ(u12 ∨ v12 ∨ w2)Z(u2, v2)

Some calculation (cf. [158]) yields

(1.23)∑

Z(u1 ∨ v1, w)Z(u2, v2) = ζ−1(u1)ζ−1(v1)ζ−1(w1)ζ(u2 ∨ v2 ∨ w2)

The right side is fully symmetric in u, v, w so all permutations of u, v, w in the leftside give the same result. In particular the permutation (u, v, w) → (v, w, u) andthe symmetry of Z transform the left side of (1.20) into its right side, and thisproves (1.20). One can check also that the Laplace pairing satisfies the couplingidentity

(1.24)∑

(u1 ∨ v1|w)(u2|v2) =∑

(u|v1 ∨ w1)(v2|w2)

It would be interesting to know the most general solution of the coupling identity.

One defines next a modified Laplace pairing via (♣) (u|v) =∑

kZ(u1, v1)(u2|v2);thus e.g.

(1.25) (u|1) = (1|u) = ε(u); (a|b) = ζ ∗ a ∨ b) + (a|b); (a|b ∨ c) = ζ(a ∨ b ∨ c)

The modified Laplace pairing also satisfies the coupling identity

(1.26)∑

(u1 ∨ v1|w)(u2|v2) =∑

(u|v1 ∨ w1)(v2|w2)

(cf. [158]). Finally one can define a renormalized circle product via (♠) uv =∑(u1|v1)u2 ∨ v2. As examples one has

(1.27) 1u = u1 = u; (u|v) = ε(uv); ab = a ∨ b + ζ(a ∨ b) + (a|b)The renormalised circle product is associative (cf. [158]) and that is considered amain result. One notes that the renormalisation group acts on the circle product(not on the elements of the algebra) even thought the action is expressed via thealgebra.

1.2. PRODUCTS AND RELATIONS TO PHYSICS. In QFT the bo-son operators commute inside a t-product and we consider now the circle productto be commutative. Then in particular a b = b a so (a|b) = (b|a). Converselyif (a|b) = (b|a) for all a, b ∈ V then u v = v u for all u, v ∈ S(V ). To see thisnote that if (a|b) = (b|a) then (u|v) = (v|u) for all u, v ∈ S(V ) (cf. (1.6)). Thenbecause of the commutativity of the symmetric product one obtains

(1.28) u v =∑

u1 ∨ v1(u2|v2) =∑

v1 ∨ u1(v2|u2) = v u

Now one defines a T map from S(V ) to S(V ) via T (1) = 1, T (a) = a for a ∈ Vand T (u ∨ v) = T (u) T (v). More explicitly T (a1 ∨ · · · an) = a1 · · · an. Thecircle product is associative and here commutative so T is well defined. The mainproperty we need is

(1.29) ∆T (u) =∑

u1 ⊗ T (u2) =∑

T (u1)⊗ u2


(cf. [158] for proof). Next one shows that the T map can be written as theexponential of an operator Σ. First define a derivation δk attached to a basis ek

of V by requiring

(1.30) δk1 = 0, δkej = δkj ; δk(u ∨ v) = (δku) ∨ v + u ∨ (δkv)

This gives a Leibnitz relation yielding δ1(e1 ∨ e2) = e2 and δ2(e1 ∨ e2) = e1. Fromthis follows δiδj = δjδi; further one notes δi(u|v) = 0. Now define the infinitesimalT map as Σ = (1/2)

∑i,j(ei|ej)δiδj . One shows that T = exp(Σ). First one needs

[Σ, a] =∑

(a|ei)δi and to see this apply Σ to an element ek ∨ u to get

(1.31) Σ(ek ∨ u) =12

∑i,j

(ei|ej)δiδj(ek ∨ u) =12

∑(ei|ej)δi(δjku + ek ∨ δju) =

=12

∑i

(ei|ej)δiu+12

∑j

(ek|ej)δju+12

∑(ei|ej)ek∨δiδju =

∑j

(ek|ej)δju+ek∨Σu

Extending this by linearity to V one obtains Σ(a∨ u) = a∨Σu +∑

(a|ej)δju andmore generally

(1.32) Σ(u ∨ v) = (Σu) ∨ v + u ∨ (Σv) +∑i,j

(ei|ej)(δiu) ∨ (δjv)

From the commutation of the derivations we obtain [Σ, δk] = 0 and [Σ, [Σ, a]] = 0;therefore the classical formula (cf. [507]) yields exp(Σ)aexp(−Σ) = a + [Σ, a] sothat

(1.33) [exp(Σ), a] =∑

(a|ei)δiexp(Σ) = exp(Σ)∑

(a|ei)δi ≡

≡ exp(Σ)(a ∨ u) = a ∨ (exp(Σ)u) + [Σ, a](exp(Σ)u)

Note also a u = a ∨ u + [Σ, a]u. From this one preceeds inductively. One hasT (1) = exp(Σ)1 = 1 and T (a) = a = exp(Σ)a since Σa = 0. Assuming theproperty T = exp(Σ) is true up to grading k take u ∈ Sk(V ) and calculate

(1.34) T (a∨u) = aT (u) = a∨T (u)+[Σ, a]T (u) = a∨eσu+[Σ, a]eΣu = eΣ(a∨u)

Thus the property is true for a ∨ u of grading k + 1.

Now one can write the T map as a sum of scalars multiplied by elements ofS(V ) in the form T (u) =

∑t(u1)u2 where t is a linear map S(V ) → C defined

recursively by t(1) = 1, t(a) = 0 for a ∈ V and

(1.35) t(u ∨ v) =∑

t(u1)t(v1)(u2|v2)

This is called the t map and the details are in [158]. The t map is well definedbecause t(u) = ε(T (u)) which can be proved by recursion. It is true for u = 1 andu = a and if true up to grading k take w = u ∨ v and one has

(1.36) ε(T (w)) =∑

t(w1)ε(w2) = t(∑

w1ε(w2)) = t(w)

A few examples are

(1.37) t(a ∨ b) = (a|b); t(a ∨ b ∨ c ∨ d) = (a|b)(c|d) + (a|c)(b|d) + (a|d)(b|c)

The general formula for t(a1 ∨ · · · ∨a2n) has (2n− 1)!! terms which can be written

(1.38) t(a1 ∨ · · · ∨ a2n) =∑

σ

n∏j=1

(aσ(j)|aσ(j+n))

where the sum is over permutations σ of (1, · · · , 2n) such that σ(1) < · · · < σ(n)and σ(j) < σ(j +n) for j = 1, · · · , n. Alternatively as a sum over all permutationsof (1, · · · , 2n) (since (a|b) = (b|a) here) one has

(1.39) t(a1 ∨ · · · ∨ a2n) =1

2nn!

∑σ

(aσ(1)|aσ(2)) · · · (aσ(2n−1)|aσ(2n))

Next one defines renormalised T maps and shows that they coincide with therenormalised t products of QFT. A renormalised T map T is a linear map S(V ) →S(V ) such that T (1) = 1, T (a) = a for a ∈ V , and T (u ∨ v)T (u)T (v). Since thecircle product is assumed commutative here the renormalised circle product is alsocommutative and the map T is well defined. Now following the proof of (1.29)one can show that ∆T (u) =

∑u1 ⊗ T (u2) and one derives two renormalisation

identities which we state without proof, namely

(1.40) T (u)T (v) =∑

Z(u1, v1)T (u2) T (v2); T (u) =∑

ζ(u1)T (u2)

The second is a second main result of [158] and corresponds to Pinter’s identity in[770]. Its importance stems from the fact that it is valid also for field theories thatare only renormalisable with an infinite number of renormalisation parameters. Aproof of the formula for T (u) goes as follows. It is true for u = 1 or u = a andv = 1 or v = b. Suppose it holds for u and v; then by definition and using therecursion hypothesis

(1.41) T (u ∨ v) = T (u)T (v) =∑

ζ(u1)ζ(v1)T (u2)T (v2)

But a u = a ∨ u + [Σ, a]u so

(1.42) T (u ∨ v) =∑

ζ(u1)ζ(v1)Z(u21, v21)T (u22) T (v22)

From the coassociativity of the coproduct and the definition (1.18) of the Z pairingone obtains

(1.43) T (u ∨ v) =∑

ζ(u1 ∨ v1)T (u2) T (v2) =∑

ζ((u ∨ v)1)T ((u ∨ v)2)

which is the required identity for u ∨ v.

Now the reasoning leading to the scalar t map can be followed exactly todefine a scalar renormalised t map as a linear map t : S(V ) → C such thatt(1) = 1, t(a) = 0 for a ∈ V and t(u ∨ v) =

∑t(u1)t(v1)(u2|v2). Consequently

t(u) = ε(T (u)) and T (u) =∑

t(u1)u2. Further T (u) =∑

ζ(u1) ⊗ T (u2) enablesone to show that t(u) =

∑ζ(u1)t(u2). As examples consider

(1.44) t(a ∨ b) = (a|b) + ζ(a ∨ b); t(a ∨ b ∨ c) = ζ(a ∨ b ∨ c)

The scalar renormalised t produce corresponds to the numerical distributions ofthe causal approach (cf. [768, 848]).

To define a normal product one starts from creation and annihilation operatorsa+

k and a−k taken as basis vectors of V + and V −. These two bases are in involution

via (a+k )∗ = a−

k and (a−k )∗ = a+

k . The creation operators commute and there is noother relation between them (similarly for the annihilation operators). Thus theHopf algebra of the creation operators is the symmetric algebra S(V +) and thatof the annihilitation operators is S(V −). One defines V = V +⊕V − (a = a+ +a−)and let P (resp. M) be the projector V → V + (resp. V → V −). Thus P (a) = a+

and M(a) = a−. There is an isomorphism φ between S(V ) and the tensor productS(V +) ⊗ S(V −) via φ(u) =

∑P (u1) ⊗ M(u2). Now S(V ) is the vector space

of normal products and one recovers the fact that a normal product puts allannihilation operators on the right of all creation operators. The isomorphismcan be defined recursively via

(1.45) P (1) = 1; M(1) = 1; P (a) = a+; M(a) = a−; φ(1) = 1⊗ 1;

φ(a) = P (a)⊗ 1 + 1⊗M(a);

φ(u ∨ v) =∑

P (u1)P (v1)⊗M(u2)M(v2);

P (u ∨ v) = P (u)P (v); M(v ∨ v) = M(u)M(v)The algebra S(V ) is graded by the number of creation and annihilation operators.Note here a subtelty. It appears that one has forgotten the operator productof elements of V + with elements of V −. One has replaced a+

k a− by a+

k ⊗ a−

and lost all information concerning the commutation of creation and annihilationoperators. But an operator with creation operators on the left and annihilationoperators on the right would not be well defined (cf. [935]). For instance a−

k a+k and

a+k a−

k +1 are equal as operators but their normal products a+k a−

k and a+k a−

k +1 aredifferent!. Thus it is not consistent to consider a normal product as obtained fromthe transformation of an operator product and one naturally loses all informationabout commutation relations in QFT. However one recalls that an antisymmetricLaplace pairing enables one to build a circle product in S(V ) which is an operatorproduct. In that sense normal products are a basic concept and operator productsare a derived concept of QFT. Thus in the present picture QFT starts with thespace S(V ) of normal products and the operator product and time ordered productare deformations of the symmetric product.

Note in QFT for scalar particles the basis ak is not indexed by k but by acontinuous variable x and (unfortunately) creation operators are denoted by φ−(x)and annihilation operators by φ+(x). This leads (cf. [507]) to a bilinear form forthe definition of operator products of the type (φ(x)|φ(y)) = iD(x − y) and abilinear form for the time ordered products as (φ(x)|φ(y)) = iGF (x− y) where formassless bosons

(1.46) D(x) = − 12π

sign(x0)δ(x2); GF (x− y) =1

4π2

1x2 + i0

are distributions.

There is a striking identity in S(V ), namely ε(u) =< 0|u|0 > which canbe shown as follows. First < 0|u|0 > is a linear map S(V ) → C. Second forelements u ∈ Sn(V ) with n > 0 one has ε(u) = 0 and < 0|u|0 >= 0 because

2. QUANTUM FIELDS AND QUANTUM GROUPS 327

u is a normal product. Finally the symmetric Hopf algebra S(V ) is connectedso all elements of S0(V ) are multiples of the unit (i.e. u = λ(u)1). But thenε(u) = λ(u)ε(1) = λ(u) so u = ε(u)1. But the vacuum is assumed to be normalizedso < 0|u|0 >= ε(u) < 0|1|0 >= ε(u) showing that the counit and the vacuumexpectation value are the same. We refer to [158] (and subsequent sections) forfurther comments on the physics.

2. QUANTUM FIELDS AND QUANTUM GROUPS

We follow now [157, 160]. First from [160] we recall that a quantum group(QG) is a quasitriangular Hopf algebra (cf. [192]). Thus we want a Hopf algebra Hwith an R matrix R ∈ H⊗H. However since Hopf algebras have a self dual naturewe can also describe a QG as a Hopf algebra with a coquasitriangular structure.This means there is an invertible linear map R : H ×H → C such that

(2.1) R(a · b, c) =∑

R(a, c1)R(b, c2); R(a, b · c) =∑

R(a1, c)R(a2, b)

For a commutative and cocommutative Hopf algebra no other condition is requiredfor R and one restricts consideration to this situation. One uses R to define atwisted product

(2.2) a b =∑

R(a1, b1)a2 · b2

This is a special case of Sweedler’s crossed product (see [923]) and given (2.1)with a cocommutative coproduct the twisted product is associative. Now considera real scalar particle with field operator

(2.3) φ(x) =∫

dk(2π)3

√2ωk

(e−ip·xa(k) + eip·xa†(k)

)where ωk =

√m2 + |k|2, p = (ωk,k), and a†(k), a(k) are the creation and anni-

hilation operators acting in a symmetric Fock space of scalar particles Fs. Oneconsiders the vector space of smoothed fields V = φ(f), f ∈ D(R4) whereφ(f) =

∫dxf(x)φ(x) (D is the standard Schwartz space). Now from V one

builds the symmetric Hopf algebra S(V ) = ⊕∞0 Sn(V ) as in Section 8.1 with

S0(V ) = C · 1, S1(V ) = V, and Sn(V ) generated by the symmetric productof n elements of V. Here one is thinking of φ(f) ∨ φ(g) ∼: φ(f)φ(g) : and 1 is theunit operator in Fs. The coproduct is defined via ∆φ(f) = φ(f) ⊗ 1 + 1 ⊗ φ(f)and extended to S(V ) as before; thus e.g.

(2.4) ∆(φ(f) ∨ φ(g)) = (φ(f) ∨ φ(g))⊗ 1 + φ(f)⊗ φ(g)+

+φ(g)⊗ φ(f) + 1⊗ (φ(f) ∨ φ(g))

(cf. (1.3)). This coproduct is cocommutative with counit ε(a) = 0 for a ∈ Sn(V )with n > 0 and ε(1) = 1; further ε(a) =< 0|a|0 > as indicated in Section 8.1. ThusS(V ) is a commutative, cocommutative, star Hopf algebra with the involutionφ(f)∗ = φ(f∗) where f∗ = f . To make S(V ) a quantum group now we canadd a coquasitriangular structure determined entirely by its value on V. Thus if

a = u1 ∨ · · · ∨ vm ∈ Sm(V ) and b = v1 ∨ · · · ∨ vn ∈ Sn(V ) then (2.1) implies thatR(a, b) = 0 if m = n and R(a, b) = perm(ui, vj) if m = n. Here

(2.5) perm(ui, vj) =∑

R(u1, vσ(1)) · · ·R(un, vσ(n))

where the sum is over all permutations σ of (1, · · · , n). The most general Poincareinvariant coquasitriangular structure on S(V ) is determined by values on V whichcan be written

(2.6) R(φ(f), φ(g)) =∫

dxdyf(x)G(x− y)g(y)

where G(x) is a Lorentz invariant distribution. Now R gives rise to a twistedproduct on S(V ) as in (2.3) and by choosing the proper G(x) this twisted productcan become the operator product or the time ordered product. Explicitly if

(2.7) G(x) = G+(x) =∫

dke−ip·x

(2π)32ωk

the twisted product is the usual operator product. If G(x) is the Feynman prop-agator

(2.8) GF (x) = i

∫d4pe−ip·x

(2π)4(p2 −m2 + iε)

then the twisted product (2.3) is the time ordered product. The proof is sketchedin [160] with the observation that (2.3) is of course Wick’s theorem as discussedin Section 8.1.

Next we look at the second paper in [160]. The first sections are mainly arepetition from [158] which we have covered in Section 8.2. We pick up the storyin Section 4 (with some repetition from [158, 160]). One takes a vector space V ofoperators from which the space of normal products S(V ) will be constructed. Theelements of V are defined as operators acting in a Fock space. For scalar bosonsone takes M = RN (N = 3 for the usual space time) and we set H = L2(M). Todefine the symmetric Fock space set H⊗n = H ⊗ · · · ⊗H and Sn acts on H⊗n via

(2.9) Sn(f1(k1)⊗ · · · fn(kn)) =1n!

∑σ∈Sn

fσ(1)(k1)⊗ · · · ⊗ fσ(n)(kn)

where Sn is the group of permutations of n elements. The symmetric Fock spaceis

(2.10) Fs(H) = ⊕∞n=0SnH⊗n; S0H

⊗0 = C; S1H⊗1 = H;

|ψ >= (ψ0, ψ1(k1), · · · , ψn(k1,k2, · · ·kn), · · · )where ψ0 ∈ C and each ψn(k1, · · · ,kn) ∈ L2(Mn) is left invariant under anypermutation of the variables. The components of |ψ > satisfy

(2.11) ‖|ψ > ‖2 =< ψ|ψ >= |ψ0|2 +∞∑1

|ψn(k1, · · · ,kn)|2dk1 · · · dkn < ∞

An element of Fs(H) for which ψn = 0 for all but finitely many n is called afinite particle vector and the set of such vectors is denoted by F0. Define now the

annihilation operator a(k) by its action in Fs(H). The action of a(k) on |ψ > isgiven by the coordinates (a(k|ψ > in the form

(2.12) (a(k)|ψ >n (k1, · · · ,kn) =√

n + 1ψn+1(k,k1, · · · ,kn)

The domain of a(k) is DS = |ψ >∈ F0, ψn ∈ S(Mn) for n > 0 where S is theSchwartz space of functions of rapid decrease. The adjoint of a(k) is not denselydefined but a†(k) is well defined as a quadratic form on DS × DS as follows. If|φ > and |ψ > belong to DS then () < φ|a†(k)ψ >=< a(k)φ|ψ >. If now|ψ >∈ DS then it can be checked that a(k)|ψ >∈ DS so we can calculate theoperator product a(k1) · a(k2)|ψ >∈ DS (the operator product is denoted by ·).By definitions and symmetry one knows that a(k1) · a(k2) = a(k2) · a(k1) whichimplies a†(k1) · a†(k2) = a†(k2) · a†(k1) but nothing is known about a(k1) · a†(k2)for example in view of the definition () which demands that all the a†(k) must beon the left of the a(k). The vectors a†(k1) · · · a†(km) · a(q1) · · · a(qn) for m,n ≥ 0generate a vector space W. The element where m = n = 0 is called 1 (unitoperator). The space W is equipped with a product (normal product) denoted by∨. If u = a†(k1) · · · a†(km) · a(q1) · · · a(qn) and v ∈ W then

(2.13) u ∨ v = v ∨ u = a†(k1) · · · a†(km) · v · a(q1) · · · a(qn)

W is a commutative and associative star algebra with unit generated by the a(k)and a†(k) with star structure a(k)∗ = a†(k). For V one takes V = φ(f), f ∈D(RN+1) where (cf. (2.3))

(2.14) φ(f) =∫

dx

∫dk

(2π)3√

2ωk

(f(x)e−ip·xa(k) + f(x)eip·xa†(k)

)(again ωk =

√m2 + |k|2 and p = (ωk,k). Note that the smoothed fields can also

be written as φ(f) =∫

dxf(x)φ(x) with φ(x) given via (2.3) (note that φ(x) ∈ V ).Now to define S(V ) in terms of operators we know S0(V ) = C1 where 1 is the unitoperator (i.e. for any |ψ >∈ Fs(H), 1|ψ >= |ψ >). Also we know S1(V ) = V . Toidentify Sn(V ) we need only determine the symmetric product from the normalproduct of creation and annihilation operators. Thus e.g.

(2.15) φ(f) ∨ φ(g) =∫

dxdx′∫

dk(2π)3

√2ωk

dk′

(2π)3√

2ωk′f(x)g(x′)×

×(e−i(p·x+p′·x′)a(k) · a(k′) + e−i(p·x−p′·x′)a†(k′) · a(k)+

+ei(p·x−p′·x′)a†(k) · a(k′) + ei(p·x+p′·x′)a†(k) · a†(k′))

These involve Fourier transforms and elementary arguments show that everythingis well defined and the product is an operator in DS . The same reasoning showsthat the symmetric product ∨ is well defined in S(V ) where the coproduct isgenerated by ∆φ(f) = φ(f)⊗1+1⊗φ(f) and the antipode is defined by s(φ(f1)∨· · · ∨ φ(fn)) = (−1)nφ(f1) ∨ · · · ∨ φ(fn). Again the counit gives the vacuumexpectation value as follows. Define the vacuum in DS by |0 >= (1, 0, · · · ) so bydefinitions a(k|0 >= 0 and < 0|a†(k) = 0. A little argument as before then gives

ε(u) =< 0|u|0 >. The star structure in S(V ) involves (λ1)∗ = λ1, φ(g)∗ = φ(g),and (u ∨ v)∗ = v∗ ∨ u∗ = u∗ ∨ v∗. This makes S(V ) a Hopf star algebra since

(2.16) ∆(φ(g)∗) = ∆φ(g) = φ(g)⊗1+1⊗φ(g) = φ(g)∗⊗1+1⊗φ(g)∗ = (∆φ(g))∗

Hence ∆u∗ =∑

u∗1 ⊗ u∗

2 for any u ∈ S(V ) and

(2.17) s(u∗) = (−1)nu∗; s(s(u)∗)∗ = (−1)ns(u∗)∗ = (u∗)∗ = u

A little recapitulation here for illustrative purposes gives the following. Thetwisted product can be defined from the Laplace pairing

(2.18) (φ(f)|φ(g))+ =∫

dxdyf(x)G+(x− y)g(y)

(cf. Section 8.2). This is denoted by φ(f) • φ(g) and one has as in Section 8.2φ(f)•φ(g) = φ(f)∨φ(g)+(φ(f)|φ(g))+. On the other hand Wick’s theorem givesφ(f) · φ(g) = φ(f) ∨ φ(g)+ < 0|φ(f) · φ(g)|0 > and the last term can be written< 0|φ(f) ·φ(g)|0 >=

∫dxdyf(x)g(y) < 0|φ(x) ·φ(y)|0 > where < 0|φ(x) ·φ(y)|0 >

is the Wightman function G+(x− y). Thus φ(f) • φ(g) = φ(f) · φ(g) as indicatedearlier.

For the time ordered product one considers the Laplace pairing

(2.19) (φ(f)|φ(g))F = −i

∫dxdyf(x)GF (x− y)g(y)

We denote by the corresponding twisted product; for example φ(f) φ(g) =φ(f) ∨ φ(g) + (φ(f)|φ(g))F . In particular it can be checked that

(2.20) < 0|φ(f) φ(g)|0 >= (φ(f)|φ(g))F = −i

∫dxdyf(x)GF (x− y)g(y)

We see that the twisted product of two operators is equal to their time orderedproduct because −iGF (x− y) =< 0|T (φ(x)φ(y))|0 >. One notes that GF (−x) =GF (x) so the Laplace coupling ( | ) is symmetric and the time ordered product iscommutative (it is also associative). One notes also

(2.21) < 0|T (φ(f1)φ(f2) · · ·φ(fn))|0 >= ε(φ(f1) φ(f2) · · · φ(fn)) =

= t(φ(f1) ∨ φ(f2) ∨ · · · ∨ φ(fn))

Next we go to [157] where interacting quantum fields are considered. Wesummarize as follows. If C is a cocommutative coalgebra with coproduct ∆′

and counit ε′ the symmetric algebra S(C) = ⊕∞0 Sn(C) can be equipped with

the structure of a bialgebra. The product is denoted here by · and the co-product is defined on S1(C) = C by ∆a = ∆′z and extended to S(C) via∆(uv) =

∑u1 · v1 ⊗ u2 · v2. The counit ε of S(C) is defined as ε′ on S1(C) = C

and extended via ε(1) = 1 with ε(u · v) = ε(u)ε(v). One checks that ∆ is coasso-ciative and cocommutative and that

∑ε(u1)u2 =

∑u1ε(u2) = u. Thus S(C) is

a commutative and cocommutative bialgebra graded as an algebra. We note that∆k is defined via ∆0a = a, ∆1a = ∆a, and ∆k+1a = (id⊗ · · · ⊗ id⊗∆)∆ka with

∆ka =∑

a1 ⊗ · · · ⊗ ak+1. A Laplace pairing is (again) defined as in (1.5) (with ∨replaced by ·). Then from these definitions one gets for uj , vj ∈ S(C)

(2.22) (u1 · · · · · uk|v1 · · · · · v) =∑ k∏

i=1

∏j=1

(uij |v

ji )

For example (u·v ·w|s·t) =∑

(u1|s1))u2|t1)(v1|s2)(v2|t2)(w1|s3)(w2|t3). A Laplacepairing is entirely determined by its value on C and it induces a twisted product on S(C) as before (cf. (1.7)). Applying the counit to both sides in (1.7) yieldsε(u v) = (u|v). Following the proofs in [160] (sketched in Section 8.1) one getsformulas such as (1.10) - (1.11), etc. and in addition

(2.23) ∆(u1 · · · uk) =∑

u11 · · · uk

1 ⊗ u12 · · · · · uk

2

This leads to the important relation

(2.24) u1 · · · uk =∑

ε(u11 · · · uk

1)u12 · · · · · uk

2

A second important identity in this context is also proved, namely

(2.25) ε(u1 · · · uk) =∑ k−1∏

i=1

k∏j=1+1

(uij−1|u

ji ) =

∑∏j>i

(uij−1|u

ji )

(the proof is in [157]).

Now one goes to interacting quantum fields. Recall the scalar fields are definedvia (2.3) and interacting fields are products of fields at the same point. Thusdefine powers of fields φn(x) as the normal product of n fields at x (i.e. φn(x) =:φ(x) · · ·φ(x) :). This is meaningful for n > 0 and φ0(x) = 1. Consider thecoalgebra C generated by φn(x) where x runs over space time and n goes from 0 to3 for a φ3 theory and from 0 to 4 for a φ4 theory. The coproduct in C is ∆φn(x) =∑n

0 φk(x) ⊗ φn−k(x) and the counit is ε(φn(x)) = δn,0. Scalar fields are bosonsso we work with the symmetric algebra S(C) with product =: uv : and ε(u) =<0|u|0 >. In S(C) the Laplace pairing is entirely determined by (φn(x)|φm(x))which itself is determined by the value of (φ(x)|φ(y) = G(x, y) if we considerφn(x) as a product of fields. More precisely (φn(x)|φm(y)) = δm,nG(x, y)(n) whereG(x, y)(n) = (1/n!)G(x, y)n. Here G ∼ G+ or GF as before. Now noting that∆k−1φn(x) =

∑φm1(x)⊗ · · · ⊗ φmk(x), with a sum over all nonnegative integers

mi such that∑k

1 mi = n one can specialize (2.22) to C as

(2.26) (: φn1(x1) · · ·φnk(xk) : | : φp1(y1) · · ·φp(y) :) =∑M

k∏i=1

∏j=1

G(xi, yj)(mij)

where the sum is over all k × matrices M of nonnegative integers mij such that∑j=1 mij = ni and

∑ki=1 mij = pj . Note now that (2.24) applied to C yields a

classical result of QFT, namely

(2.27) φ(n1)(x1)· · ·φ(nk)(xk) =n1∑

i1=0

· · ·nk∑

ik=0

< 0|φ(i1)(x1)· · ·φ(ik)(xk)|0 > ×

× : φ(n1−i1)(x1) · · ·φnk−ik)(xk) :This equation appeared in [339] but (2.24) is clearly more compact and moregeneral than (2.27). Finally specializing (2.25) to C one obtains

(2.28) < 0|φ(n1)(x1) · · · φ(nk)(xk)|0 >=∑M

k−1∏i=1

k∏j=1+1

G(xi, xj)(mij

over all symmetric k×k matrices M of nonnegative integers mij with∑k

j=1 mij =nj and mii = 0 for all i. When the twisted product is the operator product thisexpression appears in [168] but (2.28) is proved with a few lines of algebra whereasthe QFT proof is long and combinatorial. When the twisted product is the timeordered product (2.28) has a diagrammatic interpretation in the spirit of Feynman(cf. [157] for details).

3. RENORMALIZATION AND ALGEBRA

We go now to [157] and start with the physics in order to motivate the algebra.Thus let B be the bialgebra of scalar fields generated by φn(x) for x ∈ R4 andn ≥ 0 (recall φn(x) is the normal product of n fields φ(x)). The product of fieldsis the commutative normal product φn(x) · φm(x) = φm+n(x) and the coproductis

(3.1) δBφn(x) =n∑0

(nk

)φk(x)⊗ φn−k(x)

The counit is εB(φn(x)) = δn,0. One defines the symmetric algebra S(B) as the al-gebra generated by the unit operator 1 and normal products : φn1(x1) · · ·φnm(xm) :.The symmetric product is the normal product and the coproduct on S(B) is de-duced from δB via

(3.2) δ : φn1(x1) · · ·φnm(xm) :=n1∑

i1=0

· · ·nm∑

im=0

(n1

i1

)· · ·

(nm

im

)×

: φi1(x1) · · ·φim(xm) : ⊗ : φn1−im(x1) · · ·φnm−im(xm :The counit of S(B) is given by the expectation value over the vacuum, i.e. ε(a) =<0|a|0 > for a ∈ S(B).

Now for the time ordered product of a ∈ S(B) one takes T (a) =∑

t(a1)a2

where t(a) = ε(T (a)). In expressions like T (φn1(x1) · · ·φnm(xm)) the productinside T ( ) is usually considered to be the operator product. However since thefields φni(xi) commute inside T ( ) it is also possible to consider T as a mapS(B) → S(B). To stress this point one writes T (: φn1(x1) · · ·φnm(xm) :). FromT (a) =

∑t(a1)a2 and the definition of the coproduct and counit one recovers the

standard formula (cf. [168, 339])

(3.3) T (: φn1(x1) · · ·φnm(xm) :) =n1∑

i1=0

· · ·nm∑

im=0

(n1

i1

)· · ·

(nm

im

)×

< 0|T (: φi1(x1) · · ·φim(xm) :)|0 >: φn1−im(x1) · · ·φnm−im :

3. RENORMALIZATION AND ALGEBRA 333

The relation between two time ordered products T and T (cf. [788] and alsoSection 8.1) can be written (up to a convolution with test functions) as

(3.4) T (a) =∑

T (”O(b1) · · ·O(bk) :); a =: φn1(x1 · · ·φnm(xm) :

and the sum is over partitions of a (i.e. the different ways to write a as : b1 · · · bk :where bi ∈ S(B) and k is the number of blocks of the partition), while O is a mapS(B) → B such that O(: φn1(x1) · · ·φnm(xm) :) is supported on x1 = x2 = · · · =xm (the notation ∆ is used in [768] for the map O but is changed here to avoidconfusion with the coproduct). For consistency one must put T (1) = O(1) = 1and in standard QFT a single vertex is not renormalised so T (a) = T (a) = a ifa ∈ B. This enables one to show that O(a) = a if a ∈ B since if a ∈ B thereis only one partition of a, namely a, and (3.4) becomes T (a) = T (O(a)). ButO(a) ∈ B so T (O(a)) = O(a); the fact that T (a) = a implies that O(a) = a. Ifa =: φn1(x1) · · ·φnm(xm) : with m > 1 then (3.4) can be rewritten

(3.5) (a) = T (a) + T (O(a)) +′∑u

T (: O(b1) · · ·O(bk) :)

where∑′

u is the sum over all partitions except u = φn1(x1), · · · , φnm(xm) andu = a. However O(a) ∈ B and T acts as the identity on B so

(3.6) T (a) = T (a) + O(a) +′∑u

T (: O(b1) · · ·O(bk) :)

In fact one proves in [157] that if a =: φn1(x1) · · ·φnm(xm) : then O(a) =∑c(a1)a2 where c(a) = ε(O(a)) is a distribution supported on x1 = · · ·xm which

can be obtained recursively from t and t via

(3.7) c(a) = t(a)− t(a)−′∑u

∑c(b1

1) · · · c(bk1)t(: b1

2 · · · bk2 :)

We will sketch the induction procedure. If a =: φn1(x1) · · ·φnm(xm) : with m = 2there are only two partitions of a, namely u = : φn1(x1)φn2(x2) : and u =φn1(x1), φn2(x2). Thus T (a) = T (a) + O(a). We know that T (a) =

∑t(a1)a2

and T (a) =∑

t(a1)a2 so O(a) =∑

c(a1)a2 with c = t−t. Assume the proposition(3.7) is true up to m − 1 and take a =: φn1 · · ·φnm(xm) :. In (3.6) then theproposition is true for all O(bi) and thus

(3.8) T (a) = T (a) + O(a) +′∑u

T (: O(b1) · · ·O(bk) :) = T (a) + O(a)+

+′∑u

∑c(b1

1) · · · c(bk1)T (: b1

2 · · · bk2 :)

T (a) =∑

t(a1)a2 yields now(3.9)∑

t(a1)a2 =∑

t(a1)a2 + O(a) +′∑u

∑c(b1

1) · · · c(bk1)t(: b1

2 · · · bk2 :) : b1

3 · · · bk3 :


Since a =: b1 · · · bk : the factor : b13 · · · bk

3 : can be written a3 and the proposition isproved using the coassociativity of the coproduct. To prove the support propertiesof c(a) use the fact that O(a) is supported on x1 = · · · = xm so c(a) = ε(O(a)) issupported on the same set and c(a) = P (∂)δ(x2 − x1) · · · δ(xm − x1) where P (∂)is a polynomial in ∂xi

.

From the proposition one has now

(3.10) O(: φn1(x1) · · ·φnm(xm) :) =n1∑

i1=0

· · ·nm∑

im=0

(n1

i1

)· · ·

(nm

im

)×

×c(: φn1(x1) · · ·φnm(xm) :) : φn1−im(x1) · · ·φnm−im(xm) :

Because of the support properties of c this can be rewritten as

(3.11) O(: φn1(x1) · · ·φnm(xm) :) =n1∑

i1=0

· · ·nm∑

im=0

(n1

i1

)· · ·

(nm

im

)×

×c(: φn1(x1) · · ·φnm(xm) :) : φn1−im(x1) · · ·φnm−im(x1) :=n1∑

i1=0

· · ·nm∑

im=0

×

×(

n1

i1

)· · ·

(nm

im

)c(: φn1(x1) · · ·φnm(xm) :)φn1−im(x1) · · ·φnm−im(x1)

where the product in the last line is in B. Thus we can rewrite this as O(a) =∑c(a1)

∏a2.

It remains to relate the last result with the renormalisation coproduct. Bycombining (3.4) and O(a) =

∑c(a1)

∏a2 one gets

(3.12)T (a) =

∑u

T (: O(b1) · · ·O(bk) :) =∑

u

∑c(b1

1) · · · c(bk1)T (:

∏b12 · · ·

∏bk2 :)

If we compare this with the commutative renormalisation coproduct

(3.13) ∆b =∑

u

b11 · · · b

(u)1 ⊗

∏b12, · · · ,

∏b(u)2

one sees that T (a) =∑

C(a[1])T (a[2]) (there are two coproducts here to beindicated below and the notation [i] will be clarifed at that time); similarlyt(a) =

∑C(a[1])t(a[2]). In any event this shows that the renormalisation cooprod-

uct gives the same result as the Epstein-Glaser renormalisation. Further in orderto deal with renormalisation in curved space time renormalisation at a point isneeded and this is provided via

(3.14) φn(x) =n∑0

(nk

)c(φk(x))φn−k(x)

instead of T (φn(x)) = φn(x).

We go now to the beginning of [157] and deal with the algebra. We will takeB as a (not necessarily unital) bialgebra with coproduct δB and counit εB . The


product is denoted x ·y and T (B)+ is the subalgebra ⊕n≥1Tn(B). Here the gener-

ators x1⊗· · ·⊗xn are denoted by (x1, · · · , xn) and xi ∈ B and one uses to denotetensor product so that (x1, · · · , xn) (xn+1, · · · , xm+n) = (x1, · · · , xm+n). Finallywrite the product operation in T (T (B+) by juxtaposition so T k(T (B)+) is gener-ated by a1a2 · · · ak where ai ∈ T (B)+. The coproduct δ = δB and counit ε = εB

extend uniquely to a coproduct and counit on T (B) compatible with the multipli-cation of T (B), making T (B) a bialgebra. This construction ignores completelythe algebra structure of B; the bialgebra T (B) is the free bialgebra on the underly-ing coalgebra of B. Similarly the coproduct and counit of the nonunital bialgebraT (B)+ extend to define a free bialgebra structure on T (T (B)+). The coproductof both T (B) and T (T (B)+) is denoted by δ. Hence if one uses the Sweedler no-tation δ(x) =

∑x1⊗x2 for the coproduct in B then for a = (x1, · · · , xn) ∈ T b(B)

and u = a1 · · · ak ∈ T k(T (B)+) we have

(3.15) δ(a) =∑

(x11, · · · , xn

1 )⊗ (x12, · · · , xn

2 ); δ(u) =∑

a11 · · · ak

1 ⊗ a12 · · · ak

2

The counit is defined by ε(a) = εB(x1) · · · εB(xn) and ε(u) = ε(a1) · · · ε(ak).

Before giving the new coalgebra structure on T (T (B)+) some terminologyarises. A composition ρ is a (possibly empty) finite sequence of positive integers(parts of ρ). The length is (ρ) (number of parts) and |ρ| is the sum of the parts;ρ is called a composition of n if |ρ| = n. Let Cn be the set of all compositions ofn and by C the set ∪n≥0Cn of all compositions of all nonnegative integers. Forexample ρ = 1, 3, 1, 2 is a composition of 7 having length 4. The first four Cn

are C0 = e where e is the empty composition, C1 = (1), C2 = (1, 1), (2),and C3 = (1, 1, 1), (1, 2), (2, 1), (3). The total number of compositions of n is2n−1, the number of compositions of n of length k is (n− 1)!/(k− 1)!(n− k)!, andthe number of compositions of n containing α1 times the integer 1, α2 times 2,· · · ,and αn times n, with α1 + 2α2 + · · · + nαn = n is (α1 + · · · + αn)!/α1! · · ·αn!.The set C is a monoid under the operation of concatenation of sequences, i.e.(r1, · · · , rn)(rn+1, · · · , rm+n) = (r1, · · · , rn+m) and the identity of C is the emptycomposition e.

Now the refinement relation goes as follows. If ρ and σ are compositions withσ = (s1, · · · , sk) then ρ ≤ σ ⇐⇒ ρ factors in C as ρ = (ρ|gs1) · · · (ρ|σk) where(ρ|σi) is a composition of si for each i ∈ 1, · · · , k; (ρ|σi) is called the restrictionof ρ to the ith part of σ. For example if σ = (4, 5) and ρ = (1, 2, 1, 2, 2, 1) then ρ =(ρ|σ1) (ρ|σ2) where (ρ|σ1) = (1, 2, 1) is a composition of 4 and (ρ|σ2) = (2, 2, 1) isa composition of 5. Thus ρ ≤ σ and on can say that ρ is a refinement of σ. Notethat |ρ| = |σ| and (ρ) ≥ (σ) if ρ ≤ σ. If ρ ≤ σ one defines the quotient σ/ρ tobe the composition of (ρ) given by (t1, · · · , tk) where ti = ((ρ|σi)) for 1 ≤ i ≤ k.In the example above σ/ρ = (3, 3). Note that for σ ∈ Cn with (σ) = k one has(n)/σ = (k), σ/(1, 1, · · · , 1) = σ, and σ/σ = (1, 1, · · · , 1) ∈ Ck. Each of the setsCn (as well as all of C) is partially ordered by refinement. Each Cn has uniqueminimal element (1, · · · , 1) and unique maximal element (n) and these are all theminimal and maximal elements in C. The partially ordered sets Cn are actuallyBoolean algebras.


One proves a lemma now needed to prove coassociativity of the coproduct.Thus if ρ ≤ τ in C then the map σ → σ/ρ is a bijection from σ : ρ ≤ σ ≤ τ ontoγ : γ ≤ τ/ρ. To see this suppose ρ = (r1, · · · , rk) and γ = (s1, · · · , s) ≤ τ/ρ.Define γ ∈ C via

(3.16) γ = (r1 + · · ·+ rs1 , rs1+1 + · · ·+ rx1+s2 , · · · rk−s+1 + · · ·+ rk)

Then evidently ρ ≤ γ ≤ τ and the map γ → γ is inverse to the map σ → σ/ρ.

The monoid of compositions allows a grading on T (T (B)+). For all n ≥ 0 andρ = (r1, · · · , rk) in Cn let T ρ(B) be the subspace of T k(T (B)+) given by T r1(B)⊗· · ·⊗T rk(B). Then there is a direct sum decomposition T (T (B)+) = ⊕ρ∈CT ρ(B)where T ρ(B) · T τ (B) ⊆ T ρτ (B) for all ρ, τ ∈ C and 1T (T (B)) ∈ T e(B) (thusT (T (B)+) is a C-graded algebra). One uses this grading to define operations onT (T (B)+). Given a = (x1, · · · , xn) ∈ Tn(B) and ρ = (r1, · · · , rk) ∈ Cn definea|ρ ∈ T ρ(B) and a/ρ ∈ T k(B) via

(3.17) a|ρ = (x1, · · · , xr1)(xr1+1, · · · , xr1+r2) · · · (xn−rk+1, · · · , xn);

a/ρ = (x1 · · ·xr1 , xr1+1 · · ·xr1+r2 , · · · , xn−rk+1 · · ·xn)

where x1 · · ·xj is the product in B. More generally for u = a1 · · · a ∈ T σ(B) andρ ≤ σ in C one defines u|ρ ∈ T ρ(B) and u/ρ ∈ T σ/ρ(B) via

(3.18) u|ρ = a1|(ρ|σ1) · · · a(ρ|σ); u/ρ = a1/(ρ|σ1) · · · a/(ρ|σ)

EXAMPLE 3.1. Let ρ = (1, 2, 1, 2, 2, 1), σ = (3, 1, 2, 3), and τ = (4, 5) in Cso that ρ ≤ σ ≤ τ in C. Then σ/ρ = (2, 1, 1, 2), τ/ρ = (3, 3), and τ/σ = (2, 2). Ifu = (x1, x2, x3, x4)(y1, y2, y3, y4, y5) in T r(B) then

(3.19) u|ρ = (x1)(x2, x3)(x4)(y1, y2)(y3, y4)(y5) ∈ T ρ(B;

u|σ = (x1, x2, x3)(x4)(y1, y2)(y3, y4, y5) ∈ T σ(B);

u/ρ = (x1, x2 · x3, x4)(y1 · y2, y3 · y4, y5) ∈ T τ/ρ(B);

u/σ = (x1 · x2 · x3, x4)(y1 · y2, y3 · y4 · y5) ∈ T τ/σ(B);

(u|σ)/ρ = (u/ρ)|(σ/ρ) = (x1, x2 · x3)(x4)(y1 · y2)(y3 · y4, y5) ∈ T σ/ρ(B)

The last equality illustrates the lemma that for all ρ ≤ σ ≤ τ in C and u ∈ T r(B)the equalities

(3.20) (u|σ)|ρ = u|ρ; (u/ρ)/(σ/ρ) = u/σ; (u|σ)/ρ = (u/ρ)|(σ/ρ)

hold in Tσ/ρ(B).

We refer to [157] for proof of

LEMMA 3.1. For all ρ ≤ σ and u ∈ T σ(B) with free coproduct δ(u) =∑u1 ⊗ u2 one has

(3.21) δ(u/ρ) =∑

u1|ρ⊗ u2|ρ; δ(u/ρ) =∑

u1/ρ⊗ u2/ρ


Now define the coproduct ∆ on T (T (B)+) via

(3.22) ∆u =∑σ≤τ

u1|σ ⊗ u2/σ

for u ∈ T r(B) with free coproduct δ(u) =∑

u1 ⊗ u2. Here ∆ is called therenormalisation coproduct; it is an algebra map and hence is determined via ∆a =∑

σ∈Cna1|σ ⊗ a2/σ for all a ∈ Tn(B) with n ≥ 1. As an example consider

(3.23)∆x =

∑(x1)⊗ (x2); ∆(x, y) =

∑(x1)(y1)⊗ (x2, y2) +

∑(x1, y1)⊗ (x2 · y2);

∆(x, y, z) =∑

(x1)(y1)(z1)⊗ (x2, y2, z2) +∑

(x1)(y1, z1)⊗ (x2, y2 · z2)+

+∑

(x1, y1)(z1)⊗ (x2 · y2, z2) +∑

(x1, y1, z1)⊗ (x2 · y2 · z2)

The counit ε of T (T (B)+) is the algebra map to C whose restriction to T (B)+ isgiven by ε((x)) = εB(x) for x ∈ B and ε((x1, · · · , xn)) = 0 for n ≥ 1. One provesthen in [157]

THEOREM 3.1. The algebra T (T (B)+) together with the structure maps∆ and ε defined above is a bialgebra called the renormalisation algebra.

One has now two coproducts on T (T (B)+), namely δ and ∆ defined via (3.22)and to avoid confusion one uses an alternate Sweedler notation for the new co-product, namely ∆u =

∑u[1] ⊗ u[2] for u ∈ T (T (B)+). Note

(3.24)∑

(x)[1] ⊗ (x)[2] =∑

(x1)⊗ (x2)

for x ∈ B. A recursive definition of the coproduct can also be given. Thus theaction of B on itself by left multiplication extends to an action B⊗T (B) → T (B)denoted by x ⊗ a → x a via (•) x a = (x · x1, · · · , xn) for x ∈ B and a =(x1, · · · , xn). This action in turn extends to an action of B on T (T (B)+) denotedsimilarly by x⊗ u → x u in the form (••) x u = (x a1)a2 · · · ak for x ∈ B andu = a1 · · · ak ∈ T (T (B)+). The following proposition together with ∆1 = 1 ⊗ 1and ∆x =

∑(x1) ⊗ (x2) for x ∈ B determines ∆ recursively on T (B) and hence

by multiplicativity determines ∆ on all T (T (B)+).

PROPOSITION 3.1. For all a ∈ T (B) with n ≥ 1 and x ∈ B

(3.25) ∆((x) a) =∑

(x1)a[1] ⊗ (x2) a[2] +∑

(x1) a[1] ⊗ x2 a[2]

We refer to [157] for proof. One may formulate (3.25) as follows. Correspond-ing to an element x ∈ B there are 3 linear operators on T (T (B)+), namely

(3.26) Ax(u) = x u; Bx(u) = ((x) a1)a2 · · · ak; Cx(u) = (x)u

induced by left multiplication in B, T (B), and T (T (B)+) respectively. With thisnotation (3.25) takes the form

(3.27) ∆(Bx(a)) =∑

(Cx1 ⊗Bx2 + Bx1 ⊗Ax2)∆a


As a third formulation let A,B,C be the mappings from B to the set of linearoperators on T (T (B)+) respectively given by x → Ax, x → Bx, and x → Cx.Then (3.25) takes the form

(3.28) ∆(Bx(a)) = (A⊗B + B ⊗ C)(δ(x))(∆(a)

One also has

(3.29) ∆(Ax(a)) = (A⊗A)(δ(x))(∆a); ∆(Cx(a)) = (C ⊗ C)(δ(x))(∆a)

Finally one has a more explicit expression for the coproduct which is useful indefining a coproduct on S(S(B)+). Thus if a = (x1, · · · , xn) one has

(3.30) ∆a =∑

u

a11 · · · a

(u)1 ⊗ (

∏a12, · · · ,

∏a

(u)2 )

where the product a11 · · · a

(u)1 is in T (T (B)+). This is a simple rewriting of (3.22)

and u runs over the compositions of a. By a composition of a one means an elementu ∈ T (T (B)+) such that u = (a|ρ) for some ρ ∈ Cn. If the length of ρ is k one canwrite u = a1 · · · ak whre ai ∈ T (B) are called the blocks of u. Finally the lengthof u is (u) = (ρ) = k. To complete the definition of (3.30) one must still defineai1 and

∏ai2. If ai = (y1, · · · , ym) is a block then ai

1 = (y11 , · · · , ym

1 ) ∈ T (B)+ and∏ai2y

12 · · · ym

2 ∈ B.

Now assume B is a graded bialgebra which is no loss of generality since onecan always consider that all elements of B have degree zero. The grading of B

will be used to define a grading on T (T (B)+). Denote by |x| the degree of ahomogeneous element x ∈ B and by deg(a) the degree (to be defined) of a ho-mogeneous a ∈ T (T (B)+). First for T (B) the degree of 1 is zero, the degree of(x) ∈ T 1(B) is equal to the degree |x| of x ∈ B. More generally the degree of(x1, · · · , xn) ∈ T (B) is (A82) deg((x1, · · · , xn)) = |x1|+ · · ·+ |xn|+n−1. Finallyif a1, · · · , ak are homogeneoous elements of T (B)+ the degree of their productin T (T (B)+) is (A83) deg(a1 · · · ak) = deg(a1) + · · · + deg(ak). Thus the degreeis compatible with the multiplication in T (T (B)+) and one shows that it is alsocompatible with the renormalisation coproduct (cf. [157] for details). For dealingwith fermions one uses a Z2 graded algebra B and we omit this for now (cf. [157]).

When the bialgebra B is commutative one can work with the symmetric al-gebra S(S(B)+) and to define S(T (B)+) one takes quotients from T (T (B)+) viathe ideal I = u(ab− ba)v; u, v ∈ T (T (B)+), a, b ∈ T (B)+. Note that

(3.31) ∆u(ab− ba)v =∑

u[1]a[1]b[1]v[1] ⊗ u[2]a[2]b[2]v[2] −∑

u[1]b[1]a[1]v[1]⊗

⊗u[2]b[2]a[2]v[2] =∑

u[1](a[1]b[1] − b[1]a[1])v[1] ⊗ u[2]a[2]b[2]v[2]+

+∑

u[1]b[1]a[1]v[1] ⊗ u[2](a[2]b[2] − b[2]a[2])v[2]

Thus ∆I ⊂ I⊗T (T (B)+)+T (T (B)+)⊗I. Moreover ε(I) = 0 because ε(ab−ba) =0. Therefore I is a coideal and since it is also an ideal the quotient S(T (B)+) =

T (T (B)+)/I is a bialgebra which is commutative. Now one defines S(B)+ as thesubspace of T (B)+ generated as a vector space by elements

(3.32) x1, · · · , xn =∑

(xσ(1), · · · , xσ(n))

where σ runs over the permutations of n elements. The symmetric product inS(B)+ is denoted by ∨ so that x1, · · · , xn∨xn+1, · · · , xn+m = x1, · · · , xm+n.If B is commutative one shows that S(S(B)+) is a subbialgebra of S(T (B)+) withcoproduct given by (3.13) (cf. [157] for the combinatorial proof). Some examplesare given via

(3.33) ∆x =∑x1 ⊗ x2; ∆x, y =

=∑x1y1 ⊗ x2, y2+

∑x1, y1 ⊗ x2 · y2

3.1. VARIOUS ALGEBRAS. When B is the Hopf algebra of a commu-tative group there is a homomorphism between the gialgebra S(S(B)+) and theFaa di Bruno algebra. As an algebra this is the polynomial algebra generated byun for 1 ≤ n < ∞. The coproduct of un is

(3.34) ∆un =n∑

k=1

∑α

n!(u1)α1 · · · (un)αn

α1! · · ·αn!(1!)α1 · · · (n!)αn⊗ uk

The sum is over the n-tuples of nonnegative integers α = (α1, · · · , αn) such thatα + 1 + 2α2 + · · ·+ nαn = n and α1 + · · ·+ αn = k. For example

(3.35) ∆u1 = u1⊗u1; ∆u2 = u2⊗u1+u21⊗u2; ∆u3 = u3⊗u1+3u1u2⊗u2+u3

1⊗u3

It is a commutative noncocommutative bialgebra. To see the relation of this bialge-bra with the composition of functions consider formal series f(x) =

∑∞1 fn(xn/n!)

and g(x) =∑∞

1 gn(xn/n!). Define linear maps from the algebra of such formalseries to C via un(f) = fn. If one defines ∆u =

∑u1 ⊗ u2 then the terms of

f(g(x)) are

(3.36) f(g(x)) =∞∑1

un(f g)xn

n!=

∞∑1

∑un(1)(g)un(2)(f)

xn

n!

For example(3.37)df(g(x))

dx= g1f1;

d2f(g(x))dx2

= g2f1 + g21f2;

d3f(g(x))dx3

= g3f1 + 3g1g2f2 + g31f3

Following [259] it is possible to introduce a new (noncommutative) element X inthe algebra such that X,un] = un+1 and then to generate the FdB coproduct via∆u1 = u1 ⊗ u1 and ∆X = X ⊗ 1 + u1 ⊗ X. Now take the bialgebra B to be acommutative group Hopf algebra. If G is a commutative group the commutativealgebra B is the vector space generated by the elements of G and the algebra


product is induced by the product in G. The coproduct is defined by δBx = x⊗ xfor all x ∈ G. The definition (3.13) of coproduct becomes

(3.38) ∆b =∑

u

b1 · · · b(u) ⊗ ∏

b1, · · · ,∏

b(u)

The homomorphism φ between S(S(B)+) and the FdB bialgebra is given by φ(1) =1 and φ(a) = un for any a ∈ Sn(B) with n > 0. It can be established from (3.38)by noticing that the number of partitions of x1, · · · , xn with α1 blocks of size1, · · · , αn blocks of size n is n!/α1! · · ·αn!(1)α1 · · · (n!)αn .

Next comes the Pinter Hopf algebra. The bialgebras T (T (B)+) and S(S(B)+)can be turned into Hopf algebras by quotienting with an ideal. The subspaceI = (x)− εB(x)1; x ∈ B is a coideal because

(3.39) ∆((x)− εB(x)1) =∑

(x1)⊗ (x2)− εB(x)1⊗ 1 =∑

(x1)⊗ (x2)−

−∑

εB(x1)εB(x2)1⊗1 =∑

((x1)−εB(x1)1)⊗(x2)+∑

εB(x1)1⊗((x2)−εB(x2)1)

Since ε(I) = 0 the subspace I is a coideal. Therefore the space J of elementsof the form uav where u, v ∈ T (T (B)+) and a ∈ I is an ideal and a coideal soT (T (B)+)/J is a bialgebra. The action of the quotient is to replace all the (x) byε(x)1. For example

(3.40) ∆(x, y) =∑

1⊗ (x, y) +∑

(x, y)⊗ 1; ∆(x, y, z) =∑

1⊗ (x, y, z)+

+∑

(y1, z1)⊗ (x, y2 · z2) +∑

(x1, y1)⊗ (x2 · y2, z) +∑

(x, y, z)⊗ 1

More generally if a = (x1, · · · , xn) one writes

(3.41) ∆a = a⊗ 1 + 1⊗ a +′∑

a1 ⊗ a2

where∑′ involves elements a1, a2 of degrees strictly smaller that the degree of a.

Hence the antipode can be defined as in [861] and T (T (B)+)/J is a connectedHopf algebra. The same is true of S(S(B)+)/J ′ where J ′ is the subspace ofelements of the form uav with u,∈ S(S(B)+) and a ∈ x − εB(x)1; x ∈ B.

Finally consider the Connes-Moscovici algebra. If we take the same quotientof the FdB bialgebra (i.e. by letting u1 = 1) one obtains a Hopf algebra (FdBHopf algebra). In [259] there is defined a Hopf algebra related to the FdB Hopfalgebra as follows. If φ(x) = x +

∑∞2 unxn/n! one defines δn for n > 0 via

(3.42) −logφ′(x) =∞∑1

δn(xn/n!)

To calculate δn as a function of uk write φ′(x) = 1+∑∞

1 un+1xn/n! and −log(1+

z) =∑∞

1 (−1)k−1(k − 1)!zk/k!. Since the FdB formulas describe the compositionof series one can use it to write

(3.43) δn =n∑1

(−1)k−1(k − 1)!∑α

n!(u2)α2 · · · (un+1)αn

α1! · · ·αn!(1!)α1 · · · (n!)αn

4. TAU FUNCTION AND FREE FERMIONS 341

where the sum is over the n-tuples of nonnegative integers α = (α1, · · · , αn) asbefore. For example δ1 = u2, δ2 = u3 − u2

2, δ3 = u4 − 3u3u2 + 2u32, etc. Except

for the shift the relation between un and δn

the moments of a distribution and its cumulants, or between unconnected Greensfunctions and connected Greens functions. The inverse relation is obtained fromφ(x) =

∫ x

0dtexp(−

∑∞1 δn(tn/n!). Thus

(3.44) un+1 =∑

n!(δ1)α1 · · · (δn)αn)/(α1! · · ·αn!(1!)α1 · · · (n!)αn)

4. TAU FUNCTION AND FREE FERMIONS

We follow here [191, 192, 205, 455, 737, 738, 739]. For free fermions we∗

(4.1) ψn|0 >= 0 =< 0|ψ∗n (n < 0); ψ∗

n|0 >= 0 =< 0|ψn (n ≥ 0)

Here the algebra of free fermions is a Clifford algebra A over C with

(4.2) [ψm, ψn]+ = [ψ∗m, ψ∗

n]+ = 0; [ψm, ψ∗n] = δmn

An element of W = (⊕m∈ZCψm) ⊕ (⊕m∈ZCψ∗m) is called a free fermion. The

Clifford algebra has a standard representation (Fock representation) as follows.Let

(4.3) Wan = (⊕m<0Cψm)⊕ (⊕m≥0Cψ∗m); Wcr = (⊕m≥0Cψm)⊕ (⊕m<0Cψ∗

m)

Then consider the left (resp. right) A-module F = A/AWan (resp. F ∗ =WcrA/A). These are cyclic A- modules generated by the vectors |0 >= 1 modAWan (resp. by < 0| = 1 mod WcrA) with the properties given in (4.1). The Fockspaces F and F ∗ are dual with pairing defined via the vacuum expectation value< 0| · |0 > which has the properties

(4.4) < 0|1|0 >= 1; < 0|ψmψ∗m|0 >= 1 (m < 0); < 0|ψ∗

mψm|0 >= 1 (m ≥ 0);

< 0|ψmψn|0 >= (0|ψ∗mψ∗

n|0 >= 0; < 0|ψmψ∗n|0 >= 0 (m = n)

and the Wick property applies, namely

(4.5) < 0|w1 · · ·w2n+1|0 >= 0;

< 0|w1 · · ·w2n|0 >=∑

σ

sgn(σ) < 0|wσ(1)wσ(2)|0 > · · · < 0|wσ(2n−1)wσ(2n)|0 >

where wk ∈ W and σ runs over permutations such that σ(1) < σ(2), · · · , σ(2n −1) < σ(2n) and σ(1) < σ(3) < · · · < σ(2n − 1). One considers also infinitematrices (aij)i,j∈Z satisfying the condition that there is an N such that aij = 0for |i − j| > N ; these are called generalized Jacobi matrices. They form a Liealgebra with bracket [A,B] = AB − BA. The quadratic elements

∑aij : ψiψ

∗j :

are important where : ψψ∗j := ψiψ

∗j− < 0|ψiψ

∗j |0 >. These elements (with 1) span

an infinite dimensional Lie algebra g(∞) where

(4.6) [∑

aij : ψiψ∗j :,

∑bij : ψiψ

∗j :] =

∑cij : ψiψ

∗j : +c0;

cij =∑

k

aikbkj −∑

k

bikakj ; c0 =∑

i<0,j≥0

aijbji −∑

i≥0,j<0

aijbji

adopt the notation of [737] for convenience (note in [191] ψ ↔ ψ ). Thus

is the same as the relation between

The last term c0 commutes with each quadratic term. Thus the Lie algebra ofquadratic elements

∑aij : ψiψ

∗j : is different from the algebra of generalized

Jacobi matrices by virtue of the central extension c0.

Now g = exp(∑

anm : ψnψm :) is an operator belong to the formal groupcorresponding to the Lie algebra g(∞). Using (4.6) one obtains

(4.7) gψn =∑m

ψmAmng; ψ∗ng = g

∑m

Anmψ∗m

where the coefficients Anm are determined via anm. Then (4.7) implies

(4.8) [∑n∈Z

ψn ⊗ ψ∗n, g ⊗ g] = 0

This is very important and is in fact equivalent to the Hirota bilinear identity (cf.[191, 664]). One introduces now

(4.9) Hn =∞∑−∞

ψkψ∗k+n (n = 0); H(t) =

∞∑1

tnHn; H∗(t∗) =∞∑1

t∗nH−n

Here Hn ∈ g(∞) while H(t), H∗(t∗) belong to g(∞) if one restricts the numberof non-vanishing parameters tm, t∗m. For Hn we have Heisenberg algebra commuta-tion relations [Hn,Hm] = nδm+n,0. One notes also Hn|0 >= 0 =< 0|H−n (n > 0).Next one introduces the fermions

(4.10) ψ(z) =∑

k

ψkzk; ψ∗(z) =∑

k

ψ∗kz−k−1dz

Using (4.2) and (4.9) there results (ξ(t, z) =∑∞

1 tnzn)

(4.11) eH(t)ψ(z)e−H(t) = ψ(z)eξ(t,z); eH(t)ψ∗(z)eH(t) = ψ∗(z)e−ξ(t,z);

e−H∗(t∗)ψ(z)eH∗(t∗) = ψ(z)e−ξ(t∗,z−1); e−H∗(t∗)ψ∗(z)eH(t∗) = ψ∗(z)eξ(t∗,z−1)

Using

(4.12)∑n∈Z

ψn ⊗ ψ∗n = Resz=0ψ(z)⊗ ψ∗(z)

and (4.6) one arrives at

(4.13) [∑n∈Z

ψn ⊗ ψ∗n, eH(t) ⊗ eH(t)] = 0; [

∑n∈Z

ψn ⊗ ψ∗n, eH∗(t∗) ⊗ eH∗(t∗)] = 0

Therefore if g solves (4.8) then eH(t)geH∗(t∗) also solves (4.8), i.e.

(4.14) [Resz=0ψ(z)⊗ ψ∗(z), eH(t)geH∗(t∗) ⊗ eH(t)geH∗(t∗)] = 0

Next to generate some generalized tau functions one defines vacuum vectorslabelled by an integer. Thus write

(4.15) < n| =< 0|Ψ∗n; |n >= Ψn|0 >; Ψn = ψn−1 · · ·ψ1ψ0 (n > 0); Ψn =

= ψ∗n · · ·ψ∗

−2ψ∗−1 (n < 0); Ψ∗

n = ψ∗0ψ∗

1 · · ·ψ∗n−1 (n > 0); Ψ∗

n = ψ−1ψ−2 · · ·ψn

(n < 0). Now given g satisfying the bilinear identity (4.8) one constructs Kadomtsev-Petviashvili (KP) and Toda lattice (TL) tau functions via

• τKP (n, t) =< n|exp(H(t))g|n >• τTL(n, t, t∗) =< n|exp(H(t))gexp(H∗(t∗))|n >

If one fixes the variables n, t∗ then τTL becomes τKP . Next one recalls the Schurfunctions via

(4.16) sλ(t) = det(hλi−i+j(t))1≤i,j≤r

hm(t) is the elementary Schur function defined by exp(ξ(t, z)) = exp(∑∞

1 tkzk) =∑∞0 znhn(t). Another way of calculating sλ(t) follows from the formula

(4.17)< (s− k)|eH(t)ψ∗

−j1 · · ·ψ∗−jk

ψis· · ·ψi1 |0 >= (−1)j1+···jk+(k−s)(k−s+1)/2sλ(t)

where −j1 < · · · < −jk < 0 ≤ is < · · · < i1 and s − k ≥ 0. The partition λ in(4.16) is defined in a complicated way and we prefer to use (4.17).

4.1. SYMMETRIC FUNCTIONS. A few words now on symmetric func-tions are in order. Thus one defines the power sums (∗) γm = (1/m)

∑i xm

i whichis the same as the Hirota-Miwa change of variables

(4.18) mtm =∑

xmi ; mt∗m =

∑ym

i

The vector γ = (γ1, γ2, · · · ) is an analogue of the higher times in KP theory. Thecomplete symmetric functions hn are given by the generating function

(4.19)∏i≥1

(1− xiz)−1 =∑n≥1

hn(x)zn

The symmetric polynomials with rational integer coefficients in n variables form aring Λn; a ring of symmetric functions in countably many variables is denoted byΛ. If one defines hλ = hλ1hλ2 · · · for any λ then the hλ form a Z-basis of Λ. Thecomplete symmetric functions are expressed in terms of power sums via

(4.20) exp(∞∑1

γmzm) =∑n≥0

hn(γ)zn

where h0 = 1 (thus hn ∼ Schur polynomial). Suppose now that n is finite andgiven a partition λ the Schur function sλ is defined via

(4.21) sλ(x) =det(xλi+n−j

i )1≤i,j≤n

det(xn−ji )1≤i,j≤n

; det(xn−ji )1≤i,j≤n =

∏1≤i<j≤n

(xi − xj)

The Schur functions sλ(x, · · · , xn) where (λ) ≤ n form a Z-basis of Λn. Note againthat the Schur function sλ can be expressed as in (4.16) as a polynomial in the com-plete symmetric functions where n ≥ (λ) (namely sλ = det(hλi−1+j)1 ≤ i, j ≤ n).Next one defines a scalar product on Λ by requiring that for any pair of partitionsλ and µ one has

(4.22) < pλ, pµ >= δλ,µzλ; zλ =∏i≥1

imi ·mi!

where mi = mi(λ) is the number of parts of λ equal to i (here pn =∑

xmi and

the pλ = pλ1pλ2 · · · form a basis of symmetric polynomial functions with rationalcoefficients). For any pair of partitions λ and µ one has < sλ, sµ >= δλ,µ so that

the sλ form an orthonormal basis of Λ and the sλ such that |λ| = n form anorthonormal basis for the symmetric polynomials of degree n.

Consider now a function r which depends on a single variable n (integer);given a partition λ define rλ(x) =

∏i,j∈λ r(x + j − i). This is a product over

all nodes of a Young diagram (i=row, j=column); we refer to [205, 737] forpictures). The value of i − j is zero on the main diagonal and, listing the rowsvia length, the partition (3, 3, 1) will have j − i = 0, 1, 2,−1, 0, 1,−2 so rλ(x) =r(x + 2)(r(x + 1))2(r(x))2r(x− 1)r(x− 2). Given integer n and a function on thelattice r(n), n ∈ Z, define now a scalar product

(4.23) < sλ, sµ >r,n= rλ(n)δλ,µ

If ni ∈ Z are the zeros of r and k = min|n − ni| then the product (4.23) isnondegenerate on Λk. Indeed if k = n − ni > 0 the factor rλ(n) never vanishesfor partitions of length no more than k. Hence the Schur functions of k vari-abes sλ(xk), (λ) ≤ k form a basis on Λk. If n − ni = −k < 0 the factorrλ(n) never vanishes for the partitions λ : (λ′) ≤ k where λ′ is the conjugatepartition (formed by reflecting the Young diagram in the main diagonal); thensλ(xk), (λ′) ≤ k form a basis on Λk. If r is nonvanishing then the scalar prod-uct is nondegenerate on Λ∞.

Take now r(0) = 0 and set x = (1/x)r(D) where D = x(d/dx) so x · xn|x=0 =δm,nr(1)r(2) · · · r(n). Then it is proved in [737] that

(4.24) < f(x), g(x) >r,n=1n!

∆(x)f(x) ·∆(x)g(x)|x=0; ∆(x =∏

1≤i<j≤n

(xi − xj)

The proof follows from(4.25)

1n!

det(xλj+n−j)i,j=1,··· ,n · det(xµj+n−ji )i,j=1,··· ,n|x=0 = δλ,µrλ(n) =< sλ, sµ >r,n

Next one recalls the Cauchy-Littlewood formula∏

i,j(1−xiyj)−1 =∑

sλ(x)sλ(y)with Schur functions defined as in (4.21). If now γ = (γ1, γ2, · · · ) and γ∗ =(γ∗

1 , γ∗2 , · · · ) then one proves in [737] that

(4.26) exp(∞∑1

mγmγ∗m) =

∑λ

sλ(γ)sλ(γ∗)

where the sλ are constructed as in (4.16) and (4.20). We will sketch the prooflater. Consider now the pair of functions

(4.27) exp(∞∑1

mγmtm); exp(∑

mγmt∗m)

Using (4.23) and (4.26) one obtains

(4.28) < exp(∞∑1

mγmtm), exp(∞∑1

mγmt∗m) >r,n=

=∑

(λ)≤k

rλ(n)sλ(t)sλ(t∗) =∑

λ

rλ(n)sλ(t)sλ(t∗)

The series (4.28) is called a series of hypergeometric type; it is not necessarilyconvergent (cf. [737] for more on this).

Now given a partition λ = (λi) with Young diagram (λ1, λ2, · · · ) let there ber nodes (i, i). Set αi = λi − i for the number of nodes in the ith row to the rightof (i, i) and βi = λ′

i − i be the number of nodes in the ith column of λ below (i, i)(1 ≤ i ≤ r). Then α1 > α2 > · · · > αr ≥ 0 and β1 > β2 > · · · > βr ≥ 0; one writes

(4.29) λ = (α1, · · · , αr|β1, · · · , βr)

(Frobenius notation). For example given λ = (3, 3, 1) we get α1 = 3− 1 = 2, α2 =3 − 2 = 1, β1 = 3 − 1 = 2, and β2 = 2 − 2 = 0 so λ = (2, 1|2, 0) in Frobeniusnotation. Now to prove (4.26) consider the vacuum TL tau function

(4.30) < 0|exp(H(t))exp(H∗(t∗))|0 >

By the Heisenberg algebra commutation relations this is equal to exp(∑

mtmt∗m).On the other hand one can develop exp(H∗(t∗)) in Taylor series and use theexplicit form of H∗ in terms of free fermions (cf. (4.9)) and then use (4.17) to get∑

λ sλ(t)sλ(t∗) (sum over all partitions including the zero partition). Now givenλ = (i1, · · · , is|j1 − 1, · · · , js − 1) write

(4.31) |λ >= (−1)j1+···+jsψ∗−j1 · · ·ψ

∗−js

ψis· · ·ψi1 |0 >;

< λ| = (−1)j1+···+js < 0|ψ∗i1 · · ·ψ

∗is

ψ−js· · ·ψ−j1

Then

(4.32) < λ|µ >= δλ,µ, sλ(H∗)|0 >= |λ >; < 0|sλ(H) =< λ|where sλ is defined via

(4.33) sλ(t) = det(hλi−i+j(t))|1≤i,j≤(λ); exp(∞∑1

zmtm) =∞∑0

zkhk(t)

with t replaced by

(4.34) H∗ =(

H−1,H−2

2, · · · ,

H−m

m, · · ·

); H =

(H1,

H2

2, · · · ,

Hm

m· · ·

)The proof of (4.32) follows from using (4.17) and (4.26) so that exp(

∑∞1 Hmtm) =∑

λ sλ(H)sλ(t). Next one shows sλ(−A)|0 >= rλ(0)|λ > where sλ is defined via(4.33) with t replaced by(4.35)

A =(

A1,A2

2, · · · ,

Am

m, · · ·

); Ak =

∞∑−∞

ψ∗n−kψnr(n)r(n− 1) · · · r(n− k + 1)

(k = 1, 2, · · · ). The proof goes as follows. First the components of A mutu-ally commute (i.e. [Am, Ak] = 0) so apply the development exp(

∑∞1 Amtm) =∑

λ sλ(−A)sλ(t) to both sides of the right vacuum vector, i.e. exp(∑∞

1 Amtm)|0 >=∑λ sλ(t)sλ(−A)|0 >. Then use a Taylor expansion of the exponential function and

(4.35), (4.17), the definition of |λ >, and (4.32).

Recall now the definition of scalar product (4.22); it is known that one maypresent it directly via

(4.36) ∂ = (∂γ1 , (1/2)∂γ2 , · · · , (1/n)∂γn, · · · )

(cf. (*)). Then it is also known that the scalar product of polynomial functionsmay be written as

(4.37)

< f, g >= (f(∂) · g(γ))|γ=0; < γn, γm >= (1/n)δn,m; < pn, pm >= nδn,m

(which is a particular case of (4.22). One proves then in [737] that with the scalarproduct (4.37) one has

(4.38) < f, g >=< 0|f(H)g(H∗)|0 >

where H, H∗ are as in (4.34). This follows from the fact that higher times andderivatives with respect to higher times give a realization of the Heisenberg algebraHkHm −HmHk = kδk+m,0 and that

(4.39) ∂γk·mγm −mγm · ∂γk

= kδk+m,0; ∂γm· 1 = 0

(i.e. the free term of ∂γmis zero) and

(4.40) Hm|n >= 0 (m > 0); < n|Hm = 0 (m < 0)

Now consider the deformed scalar product < sµ, sλ >r,n= rλ(n)δµ,λ. Each polyno-mial function is a linear combination of Schur functions and one has the followingrealization of the deformed scalar product, namely

(4.41) < f, g >r,n=< n|f(H)g(−A)|N >

where H and A are determined as in (4.9), (4.34), (4.35). Now for the collectionof independent variables t∗ = (t∗1, t

∗2, · · · ) one writes A(t∗) =

∑∞1 t∗mAm with Am

as in (4.33). Using the explicit form of Am, (4.33), and (4.2) one obtains(4.42)

eA(t∗)ψ(z)e−A(t∗) = e−ξr(t∗,z−1) · ψ(z); eA(t∗)ψ∗(z)e−A(t∗) = eξr′ (t∗,z−1) · ψ∗(z)

where the operators

(4.43) ξr(t∗, z−1) =∞∑1

tm

(1zr(D)

)m

; D = zd

dz; r′(D) = r(−D)

act on functions of z on the right side according to the rule r(D)·zn = r(n)zn. Theexponents in (4.42) - (4.43) are considered as their Taylor series. Next using (4.42)- (4.43) and the fact that inside Resz the operator (1/z)r(D) is the conjugate of(1/z)r(−D) = (1/z)r′(D) one gets

(4.44) [Resz=0ψ(z)⊗ ψ∗(z), eA(t∗) ⊗ eA(t∗)] = 0

Now it is natural to consider the following tau function

(4.45) τr(n, t, t∗) =< n|eH(t)e−A(t∗)|n >

Using the fermionic realization of the scalar product (4.41) one gets

(4.46) τr(n, t, t∗) =< exp(∞∑1

mtmγm), exp(∞∑1

mt∗mγm) >r,n

Then due to (4.28) one obtains

(4.47) τr(n, t, t∗) =∑

λ

rλ(n)sλ(t)sλ(t∗)

Convergence questions are ignored here. The variables n, t play the role of higherKP times and t∗ is a collection of group times for a commuting subalgebra of addi-tional symmetries of KP (cf. [191, 205]). From another point of view (4.47) is atau function of a two dimensional Toda lattice with two sets of continuous variablest, t∗ and one discrete variable n. Defining r′(n) = r(−n) one has properties

(4.48) τr(n, t, t∗) = τr(n, t∗, t); τr′(−n,−t,−t∗) = τr(n, t, t∗)

Also τr(n, t, t∗) does not change if tm → amtm, t∗m → a−mt∗m, m = 1, 2, · · · and itdoes change (?) if tm → amtm, m = 1, 2, · · · , and r(n) → a−1r(n). (4.48) followsfrom the relations r′λ(n) = rλ′(−n) and sλ(t) = (−1)|λ|sλ′(−t). One should notealso that (4.47) can be viewed as a result of the action of additional symmetrieson the vacuum tau function (cf. [191, 205] for terminology).

4.2. PSDO ON A CIRCLE. Given functions r, r the operators(4.49)

Am = −∞∑−∞

r(n) · · · r(n−m+1)ψnψ∗n−m; Am =

∞∑−∞

r(n+1) · · · r(n+m)ψnψ∗n+m

belong to the Lie algebra of pseudodifferential operators (PSDO) on a circle withcentral extension. These operators may be written in the form(4.50)

Am =1

2πi

∮: ψ∗(z)

(1zr(D)

)m

· ψ(z) :; Am = − 12πi

∮: ψ∗(z)(r(D)z)m · ψ(z) :

(recall D = z(d/dz)). The action is given via r(D) · zn = r(n)zn and one works inthe punctured disc 0 < |z| < 1. Central extensions of the Lie algebra of generators(4.49) are described by the formulae

(4.51) ωn(Am, Ak) = δmkr(n + m− 1) · · · r(n)r(n) · · · r(n−m + 1)

with ωn−ωn′ ∼ 0. The extensions differing by the choice of n are cohomologicallydescribed. The choice corresponds to a choice of normal ordering : : which maybe chosen in a different manner via : a := a− < n|a|n > (n is an integer).When r, r are polynomials the operators [(1/z)r(D)]m and (r(D)z)m belong tothe W∞ algebra while the fermionic operators (4.35) and (4.49) belong to thecentral extension W∞ (cf. [579]). Here one is interested in calculating the vacuumexpectation values of different products of operators of the type exp(

∑Amt∗m)

and exp(∑

Amtm) and for partitions λ = (i1, · · · , is|j1 − 1, · · · , js − 1) and µ =(i1, · · · ir|j1 − 1, · · · , jr − 1) satisfying µ ⊆ λ one has

(4.52) < 0|ψ∗i1· · ·ψ∗

irψ−j1

· · ·ψj1eA(t∗)ψ∗

j1 · · ·ψ∗js

ψis· · ·ψi1 |0 >=

= (−1)j1+···+jr+j1+···+jssλ/µ(t∗)rλ/µ(0)

(4.53) < 0|ψ∗i1 · · ·ψis

ψ−js· · ·ψj1e

A(t)ψ∗j1· · ·ψ∗

−jrψir· · ·ψi1

|0 >=

= (−1)j1+···+jr+j1+···+jssλ/µ(t)rλ/µ(0)where sλ/µ(t) is a skew Schur function

(4.54) sλ/µ(t) = det(hλi−µj−i+j(t)); rλ/µ =

=∏

i,j∈λ/µ

r(n + j − i); rλ/µ(n) =∏

i,j∈λ/µ

r(n + j − 1)

The proof goes via a development of exp(A) = 1+A+· · · and exp(A) = 1+A+· · ·and the direct evaluation of vacuum expectations (4.52) and (4.53) along with adeterminant formula for the skew Schur function.

Now introduce vectors

(4.55) |λ, n >= (−1)j1+···+jsψ∗−j1+n, · · · , ψ∗

−js+nψis+n · · ·ψi1+n|n >;

< λ, n| = (−1)j1+···js < n|ψ∗i1+n, · · ·ψ∗

is+nψ−js+n · · ·ψ−j1+n

We see that < λ, n|µ, m >= δmnδλµ One notes from (4.10) that

(4.56)∮· · ·

∮ψ(z1) · · ·ψ(zn)|0 >< 0|ψ∗(zn) · · ·ψ∗(z1) =

∑λ,(λ)≤n

|λ, n >< λ, n|

which projects any state to the component Fn of the Fock space F = ⊕n∈ZFn.Next using (4.36) one obtains

(4.57) |λ, n >= sλ(H∗)|n >; < λ, n| =< n|sλ(H);

sλ(−A)|n >= rλ(n)sλ(H∗)|n >; < n|sλ(A) = r(n) < n|sλ(H); A =

(An

n

)where Am, Am are given via (4.31) and (4.49). One has then the following devel-opments

(4.58) e−A(t∗) =∑n∈Z

∑µ⊆λ

|µ, n > sλ/µ(t∗)rλ/µ(n) < λ, n|;

eA(t) =∑n∈Z

∑µ⊆λ

|λ, n > sλ/µ(t)rλ/µ(n) < µ, n|

where sλ/µ is the skew Schur function (4.54) and r, r are also defined in (4.54).The proof follows from the relation < sλ/µ, sν >=< sλ, sµsν >. It follows from(4.58) that

(4.59) sν(−A) =∑n∈Z

∑µ⊆λ

Cλµνrλ/µ(n)|µ, n >< λ, n|;

5. INTERTWINING 349

sν(A) =∑n∈Z

∑µ⊆λ

Cλµν rλ/µ(n)|λ, n >< µ, n|

Some calculation using < λ, n|µ, m >= δmnδλµ leads then to

(4.60) τg(n, t, t∗) =< n|exp(H(t))gexp(H(t∗))|n >=∑λ,µ

sλ(t)gλµ(n)sµ(t∗);

gλµ =< λ, n|g|µ, n >

Consequently one has the property

(4.61) (g1g2)λµ(n) =∑

ν

(g1)λν(n)(g2)νµ(n)

Hence introduce the “integrable” scalar product < sλ, sµ >g,n= gλµ(n) (generallydegenerate). This leads to

(4.62) < τg1(n1, t, γ), τg2(n2, γ, t∗) >g,n= τg3(0, t, t∗);

g3(k) = g1(n1 + k)g(n + k)g2(n2 + k)

and one actually takes here gλµ(n) = δλµrλ(n). There is much much more in [737]but we stop here for the moment.

5. INTERTWINING

We extract here from [192]. Intertwining is important in group representationtheory and we indicate some aspects of this in quantum group situations. From[191, 192, 287, 525] consider classical KP/Toda. Vertex operators are (ξ(t, z) =∑∞

1 tkzk and ξ(∂, z−1) =∑∞

1 z−k∂k/k)

(5.1) X(z) = eξ(t,z)e−ξ(∂,z−1); X∗(z) =−ξ(t,z) eξ(∂,z−1);

X∗(λ)X(µ) =λ

λ− µX(λ, µ, t); X(λ)X∗(µ) =

λ

λ− µX(µ, λ, t)

X(z, ζ, t) = e∑∞

1 (ζk−zk)e∑∞

1 (z−k−ζ−k)∂k/k =∞∑0

(ζ − z)m

m!

∞∑p=−∞

z−p−mWmp

The wave functions are defined as usual in terms of the tau function via

(5.2) ψ(t, z) =X(z)τ(t)

τ(t); ψ∗(t, z) =

X∗(z)τ(t)τ(t)

ψ∗(t, λ)ψ(t, µ) =1

µ− λ∂

(X(λ, µ, t)τ(t)

τ(t)

)Now consider the free fermion operators

(5.3) [ψn, ψm]+ = 0 = [ψ∗m, ψ∗

n]+; [ψm, ψ∗n]+ = δmn

with (|vac >∼ |0 > and < 0|vac ∼< 0|)(1) ψn|0 >= 0 =< 0|ψ∗

n (n < 0)(2) ψ∗

n|0 >= 0 =< 0|ψn (n ≥ 0)

Vacuum expectation values are defined via

(5.4) < 1 >= 1; < ψiψ∗j >= δij− < ψ∗

j ψi >=

= 1 (i = j < 0)0 otherwise

One expresses normal ordering via

(5.5) : ψiψ∗j := ψiψ

∗j− < ψiψ

∗j >; H(t) =

∞∑1

tk∑Z

ψnψ∗k+n

and we see directly that H(t)|0 >= 0 while < 0|H(t) = 0. The noncommutativealgebra generated by ψn, ψ∗

m is denoted by A and one writes V = ⊕ZCψn andV ∗ = ⊕ZCψ∗

n with W = V ⊕ V ∗. The left (resp. right) module with cyclicvector |0 > (resp. < 0|) is called a left (resp. right) Fock space on which on hasrepresentations of A via #1, 2 above. The vacuum expectation values give a Cbilinear pairing

(5.6) < 0|A⊗ A|0 >→ C; < 0|a1 ⊗ a2|0 >→< 0|a1a2|0 >

One denotes by G(V, V ∗) the Clifford group characterized via gψn =∑

ψmgamn

and ψ∗ng =

∑gψ∗

manm. Tau functions of KP are parametrized by G(V, V ∗) orbitsof |0 > for example and such orbits (modulo constant multiples) can be identifiedwith an infinite dimensional Grassmann manifold (UGM). The t-evolution of anoperator a ∈ A is defined as a(t) = exp[H(t)]aexp[−H(t)].

We note that quadratic operators ψmψ∗n satisfy

(5.7) [ψmψ∗n, ψpψ

∗q ] = δnpψmψ∗

q − δmqψpψ∗n

and (with the element 1) these span an infinite dimensional Lie algebra g(V, V ∗)whose corresponding group is G(V, V ∗). Then exp[H(t)] belongs to the formalcompletion of G(V, V ∗). One writes further Λ ∼ (δm+1,n)m,n∈Z and recalls theSchur polynomials are defined via

(5.8) exp

( ∞∑1

tkzk

)=

∞∑0

s(t)z

It follows that (via a(t) = exp[H(t)]aexp[−H(t)])

(5.9) ψn(t) =∑

ψn−s(t); ψ∗n(t) =

∑ψ∗

n+s(−t)

and for tau functions one has for v = g|0 > (recall H(t)|0 >= 0)

(5.10) τ(t, v) =< 0|eH(t)g|0 >

This exhibits the context in which tau functions eventually can be regarded interms of matrix elements in the representation theory of say G(V, V ∗) (some detailsbelow). To develop this one defines degree (or charge) via

(5.11) deg(ψn) = 1; deg(ψ∗n) = −1

5. INTERTWINING 351

so vectors ψ∗m1· · ·ψ∗

mkψnk

· · ·ψn1 |0 > with m1 < · · · < mk < 0 ≤ nk < · · ·n1

contribute a basis of A(0)|0 >. Similarly for charge n one specifies vectors

(5.12) < n| =

⎧⎨⎩ < 0|ψ−1 · · ·ψ−n (n < 0)< 0| (n = 0)

< 0|ψ∗0 · · ·ψ∗

n−1 (n > 0)

Putting in a bookkeeping parameter z one has an isomorphism

(5.13) i : A|0 >→ C[t1, t2, · · · ; z, z−1]; a|0 >→∑Z

< m|eH(t)a|0 > zm

This leads to the action of A on C[ti; z, z−1] via differential operators. Thus write

(5.14) ψ(z) =∑

ψnzn; ψ∗(z) =∑

ψ∗nz−n

and one checks that t-evolution is diagonalizable via

(5.15) eH(t)ψ(z)e−H(t) = eξ(t,z)ψ(z); eH(t)ψ∗(z)eH(t) = e−ξ(t,z)ψ∗(z)

It will then follow (nontrivially) that

(5.16) < m|eH(t)ψ(z) = zm−1X(z) < m− 1|eH(t);

< m|eH(t)ψ∗(z) = z−mX∗(z) < m + 1|eH(t)

This is actually a somewhat profound result whose clearest proof involves Wick’stheorem and we refer to [191, 205] for details. This leads to the fundamentalHirota bilinear identity

(5.17)∮

dzψ(t, z)ψ∗(t′, z) = 0

via free fermion arguments which can be written out as

(5.18)∑j∈Z

sj(2yi)sj+1(−∂y)τg(xi + yi)τg(xi − yi) = 0

in an obvious notation and this is KP theory.

Group theory underlies classical integrable systems and there are some differ-ent group structures for the same integrable system. Some of the groups act inthe space of solutions to the integrable hierarchy and others can act on just thespace time variables of the equations. In order to quantize an integrable systemone needs only replace the first type of group structures by their quantum counter-parts. However the groups acting in the space time still remain classical even forquantum systems. Now for KP elements of the form g =: exp

∑mn amnψ∗

mψn :are in 1-1 correspondence with solutions to the KP hierarchy and the group actingin the space of solutions is GL(∞). The same group acts in the Toda hierarchywith two infinite sets of times tk and tk with tau functions given by

(5.19) τn(t, t|g) =< n|eH(t)geH(t)|n >

< n|g|n >

where H(t) =∑

H−k tk. To extend KP/Toda one looks first at GL(∞) as a Hopfalgebra with ∆(g) = g ⊗ g with

(5.20) gψig−1 =

∑Rikψk; gψ∗

i g−1 =∑

ψ∗kR−1

ki

This means that the fermions are intertwining operators which intertwine thefundamental representations of GL(∞). For g(∞) each vertex on the Dynkindiagram (∞ in both directions) corresponds to a fundamental representation Fn

with arbitrary fixed origin n = 0; here ψ∗k|Fn >= 0 for k ≥ n and ψk|Fn >= 0

for k < n. The relation (5.20) implies that∑

ψi ⊗ ψ∗i commutes with g and leads

to the Hirota bilinear identities. Now for the general case (following [403, 404,548, 549, 566, 656, 663, 664, 952]) take a highest weight representation λof a Lie algebra g with UEA U(g). The tau function is defined as (cf. (5.10) -τ = τλ(t, t|g))

(5.21) τ =< 0|∏k

etkT k−g

∏i

etT i+ |0 >λ=

∑< n|g|m >λ

∏ tnii t

mj

j

ni!mj !

where the vacuum state means the highest weight vector, T k± are the generators of

the corresonding Borel subalgebras of g, and the exponentials are supposed to benormal ordered in some fashion. One then proceeds as in the classical situation.For quantization one replaces the group by the corresponding quantum group andrepeats the procedures but with the following new features:

• The tau function is no longer commutative. It is defined as in (5.21) withexponentials replaced by quantum exponentials.

• One will need to distinguish between left and right intertwiners.• The counterpart of the group element g defined by ∆(g) = g ⊗ g does

not belong to the universal enveloping algebra (UEA) Uq(g) but ratherto Uq(g)⊗U∗

q (g) where U∗q (g) = Aq(G) is the algebra of functions on the

quantum group.Thus the tau function (5.21) is the average of an element from Uq(g)⊗Aq(G) oversome representation of Uq(g) and hence belongs to Aq(G) (and is consequentlynoncommutative).

More generally one considers a universal enveloping algebra (UEA) U(g) and aVerma module V of this algebra. A tau function is then defined to be a generatingfunction of matrix elements < k|g|n >V of the form (we think of the q-deformedtheory from the outset here)

(5.22) τV (t, t|g) = V < 0|∏α>0

eq(tαTα)g∏α>0

eq(tαT−α)|0 >V =

=∑

kα≥0,nα≥0

∏α>0

tkαα tnα

α

[kα]![nα]!V < kα|g|nα >V

where [n] = (qn − q−n)/(q − q−1) with [n]! = [n][n − 1] · · · [1] and eq(x) =∑n≥0(x

n/[n]!), etc. The T±α are generators of ± maximal nilpotent subalge-bras N(g) and N(g) with a suitable ordering of positive roots α while tα andtα ∼ t−α are associated times. A vacuum state is annihilated by Tα for α > 0

5. INTERTWINING 353

(i.e. Tα|0 >= 0). The Verma module is V = |nα >V =∏

α>0 Tnα−α|0 >V . Exceptfor special circumstances all α ∈ N(g) are involved and since not all Tα are com-mutative the tau function has nothing a priori to do with Hamiltonian integrablesystems. Sometimes (e.g. for fundamental representations of s(n)) the systemof bilinear equations obtained via intertwining can be reduced to one involving asmaller number of time variables (e.g. rank(g)) and this returns one to the field ofHamiltonian integrable systems. One works with four Verma modules V, V , V ′, V ′;given V, V ′ every allowed choice of V , V ′ provides a separate set of bilinear identi-ties. The starting point is (A): Embed V into V ⊗W where W is some irreduciblefinite dimensional representation of g (there are only a finite number of possibleV ). One defines a right vertex operator (or intertwining operator) of type W as ahomomorphism of g modules ER : V → V ⊗W . This intertwining operator canbe explicitly continued to the whole representation once it is constructed for itsvacuum (highest weight) state

(5.23) V = |nα >V =∏α>0

(∆T−α)nα |0 >V

where

(5.24) |0 >V =

⎛⎝ ∑(pα,iα)

A(pα, iα)

(∏α>0

(T−α)pα ⊗ (T−α)iα

)⎞⎠ |0 >V ⊗|0 >W

Then every |nα >V is a finite sum of |mα >V with coefficients in W. Next (B):Take another triple defining a left vertex operator E′

L : V ′ → W ′ ⊗ V ′ such thatW ⊗W ′ contains a unit representation of g with projection π : W ⊗ W ′ → I.Using π one can build a new intertwining operator

(5.25) Γ : V ⊗ V ′ E⊗E′;→ V ⊗W ⊗W ′ ⊗ V ′ I⊗π⊗I→ V ⊗ V ′

such that () Γ(g⊗g) = (g⊗g)Γ for any group element g such that ∆(g) = g⊗g(() is in fact an algebraic form of the bilinear identities - one can use (5.22)to average () with the evolution exponentials over the enveloping algebra andgroup elements can be constructed using the universal T operator). Finally (C):One looks at a matrix element of () between four states, namely(5.26)

V ′ < k′| V < k|(g ⊗ g)Γ|n >V |n′ >V ′= V ′ < k′| V < k|Γ(g ⊗ g)|n >V |n

′ >V ′

and rewrites this suitably.

In terms of functions the action of Γ can be presented via

(5.27) Γ|n >λ |n′ >λ′=

∑,′| >λ ′ >λ′ Γ(, ′, n, n′)

(⊗ omitted) so (5.26) becomes

(5.28)∑

m,m′Γ(k, k′|m,m′)

‖k‖2λ‖k′‖2λ′

‖m‖2λ‖m′‖2

λ′< m|g|n >λ< m′|g|n′ >λ′=

=∑,′

< k|g| >λ< k′|g|′ >λ′ Γ(, ′|n, n′)

To rewrite this as a differential or difference expression one needs to use formulaslike (τ = τV (t, t|g))

(5.29) τ =∑

m,m∈V

sVm,m(t, t) < m|g|m >V ; τV (t, t|g) =< 0V |U(t)gU(t)|0V >

which will give a generating function for identities (making use of explicit formsfor Γ(, ′|n, n′) that arise in a group theoretic framework - Clebsch-Gordon coef-ficients, etc.).

REMARK 8.5.1. Examples abound (see e.g. [403, 404, 548, 549, 566,663, 664]) and the theory can be made very general and abstract. The factthat matrix elements for group representations along with intertwining is impor-tant is not new or surprising. What is interesting is the fact that group theoryand intertwining leads to Hirota type formulas as in KP/Toda with their accom-panying differential or q-difference equations involving tau functions as generat-ing functions for the matrix elements. One knows also of course that specialfunctions and q-special functions arise as matrix elements in the representationtheory of groups and quantum groups with associated differential or q-differenceoperators whose intertwining corresponds to classical transmutation theory as in[202, 205, 208, 209]. However from (5.21) - (5.22) for example one sees that thet variables arise from Borel subalgebras of g and it is the coefficients < n|g|m >which give rise to variables x ∼ R for the radial part of a Casimir operator (cf.[210]).

With a view toward closer relations with operator transmutation we go to[237] and consider s(2) defined via [e, f ] = h, [h, e] = 2e, [h, f ] = −2f withquadratic Casimir C = (ef + fe)+ (1/2)h2 = 2fe+h+(1/2)h2. One looks at theprincipal series of representations with space Vλ and algebra actions

(5.30) e = ∂x, h = −2x∂x + λ, f = −x2∂x + λx

the vacua correspond to constants via highest weight vectors e|0 >= 0 with h|0 >=λ|0 >. A Whittaker vector |w >µ

λ∈ Vλ with e|w >µλ= µ|w >µ

λ is given via

(5.31) |w >µλ= exp(µx) =

∑[µnfn/n!(λ, n)]|0 >

where (λ, n) = λ(λ− 1) · · · (λ− n + 1). A dual Whittaker vector is then

(5.32) µλ < w| =

∑λ < 0|[µnen/n!(λ, n)] = x−λ−2exp(−µx)

and one defines a Whittaker function W = WµL,µR

λ (φ) via(5.33)

Wλ = µL

λ < w|eφh|w >µR

λ = Wλ(φ) = 2e(λ+1)φ

(√µL

µR

)−(λ+1)

Kλ+1(2√

µLµRe−φ)

5. INTERTWINING 355

(Macdonald function Kλ). One has then

(5.34)(

12∂2

φ + ∂φ − 2µLµRe−2φ

)Wλ(φ) =

(12λ2 + λ

)Wλ(φ)

Raising and lowering operators are obtained via intertwining of Vλ spaces

(5.35) Wλ+1 = eφ ∂φ + λ + 22(λ + 1)

Wλ; µLWλ−1 =λ

µReφ λ− ∂φ

2Wλ

Similarly, using intertwiners Vλ+1 ⊗ Vν+1 → Vλ ⊗ Vν one gets bilinear relationsleading to nonlinear Hirota type equations

(5.36)∂φ + λ + 22(λ + 1)

WλµR

2 (1− ∂φ)ν + 1

Wν +µR

1

λ + 1Wλ

(∂φ + λ + 2)ν + 1

(1 + ∂φ)Wν =

= −Wλ+1µ + 1− ∂φ

2µL2

∂φWµ+1 +λ + 1− ∂φ

2µL1

Wλ+1∂φWν+1

Product formulas are also obtained and all this extends to quantum group sit-uations. Now, operator transmutation as envisioned in Section 1 would involvesay P ∼ Cλ = (1/2)∂2

φ + ∂φ − 2µLµRexp(−2φ) and Q ∼ some other operator inVλ with eigenvalue (1/2)λ2 + λ (which could be arranged by scaling). The grouptheory makes no obvious entry and in fact one is much better off doing operatortransmutation in a distribution context such as D′

φ where differential equationsinvolving Cλ for Wλ would still prevail (a Hilbert space framework is restrictive).There may be some possible use of Hirota type formulas however in studying oper-ator transmutation of “interesting” differential operators such as Cλ but this is notclear. We have mainly given in this Section a glimpse of quantum transmutations= intertwinings in the group theory context with many relations to integrable sys-tems; this seems to emphasize a deep correspondence between group theory andintegrability.

REMARK 8.5.2. Some perspective can perhaps be gained as follows. Ex-tracting here from [205] let us specify and expand now some facts to be developedfurther. The classical Wick’s theorem from say [935] is stated as: The time or-dered product of n free fields Ai equals the sum of normal ordered products of allpossible partial and complete contractions. Thus one stipulates that a contractionis A1A2 =< 0|T (A1A2)|0 > satisfying the equation : A1A2A3A4 := A2A4 : A1A3 :.Setting Cij for the operator contracting Ai and Aj one has e.g.

(5.37) T (A1A2A3) =: A1A2A3 : + : (C12 + C13 + C23)A1A2A3 :=

=: A1A2A3 : +A1A2A3 + A1A3A2 + A2A3A1

One finds also for Ai = A+i + A−

i (+ ∼ creation)

(5.38) A1A2 = [A−1 , A+

2 ] (t1 > t2); A1A2 = [A−2 , A+

1 ] (t2 > t1)

where A1A2 = T (A1A2)− : A1A2 : can serve as a definition. Comparing to Section8.1 we have (a b ∼ ab)

(5.39) a•b• (or ab) = (a|b) ∼ ab; a b ∼ T (ab) (or ab); a ∨ b ∼: ab :

Note also from (4.5)(5.40)< 0|w1 · · ·w2n|0 >=

∑σ

sgn(σ) < 0|(wσ(j)wσ(j+1)|0 > · · · < 0|wσ(2n−1)wσ(2n)|0 >

where σ(1) < σ(2), · · · , σ(2n − 1) < σ(2n) and σ(1) < σ(3) < · · · < σ(2n − 1).Now the proof sketched in [935] (for W (A1, · · · , An) the sum of all normal orderedpartial and complete contractions) involved

(5.41) T (A1 · · ·An) = A1T (A2 · · ·An) = A1W (A2, · · · , An) =

= A+1 W + WA−

1 + [A−1 ,W ] = W (A1, · · · , An)

since [A−1 ,W ] can be reduced to a sum of brackets as in (5.38). Next in (1.13) one

looks at

(5.42) : a1 · · · an : b =: a1 · · · anb : +n∑

j=1

a•j b

• : a1 · · · aj−1aj+1 · · · an :

which corresponds to a•j b

• = (aj |b) = ajb and ub = u b. Note

(5.43)

(a1 ∨ · · · ∨ an)b = a1 ∨ · · · ∨ an ∨ b +∑n

1 ajba1 ∨ · · · ∨ aj−1 ∨ aj+1 ∨ · · · ∨ an

Here u b ∼ T (ub). One compares also with (1.35), (1.38), and (1.39), Thus foru = a1 ∨ · · · ∨ an (cf. (1.4))(5.44)

T (u) =∑

t(u1)u2; ∆(u) = u⊗1+1⊗u+2n−1∑p=1

∑σ

aσ(1)∨· · · aσ(p)⊗aσ(p+1)∨· · ·∨aσ(2n)

(5.45) T (u) = t(u) + u +∑∑

t(aσ(1) ∨ · · · ∨ aσ(p))aσ(p+1) ∨ · · · ∨ aσ(2n)

Recall t(1) = 1, t(a) = 0 for a ∈ V and t(u ∨ v) =∑

t(u1)t(v1)(u2|v2). Hence inparticular

(5.46) t(a ∨ b ∨ c) =∑

t(a1 ∨ b1)t(c1)(a2 ∨ b2|c2) = 0

and we recall the equivalence of (1.38) and (1.39) (for ( | ) symmetric), namely

(5.47) t(a1 ∨ · · · ∨ a2n) =∑

σ

n∏j=1

(aσ(j)|aσ(j+n)) ≡

≡ 12nn!

∑σ

(aσ(1)|aσ(2) · · · (aσ(2n−1)|aσ(2n))

We recall also that Wick a la (5.39) implies (5.16) or

(5.48) < m|eH(t)ψ(z) = zm−1X(z) < m− 1|eH(t);

< m|eH(t)ψ∗(z) = z−mX∗(z) < m + 1|eH(t)

5. INTERTWINING 357

REMARK 8.5.3

also [45, 146, 278, 339, 356, 368, 369, 489, 582] andSection 8.6 to follow).

REMARK 8.5.4. Tau functions may provide new perspective relative totry to write down a tau

function for operator objects u, v based on ai ∈ W where W ∼ W+ ⊕W− say(e.g. W+ ∼ Wcr and W− ∼ Wan as in Section 8.4). One could look for analogues∑∞

1 tkHk withHk ∼

∑∞−∞ ψnψ∗

k+n), etc. or analogues of the vertex operators X, etc. of Section8.6, and of the Hirota equations and one could also consider Hopf algebra ideasfor free fermions. Consider normal order a ∨ b and operator product (no timeorder is needed momentarily) and ab ∼ ab ∼ (a|b) and look at some formulasfrom preceeding sections(5.49)a ∨ b ∼: ab :; Wan = ⊕m<0Cψm ⊕⊕m≥0Cψ∗

m;Wcr = ⊕m≥0Cψm ⊕⊕m<0Cψ∗m;

ψn|0 >= 0 (n < 0); ψ∗n|0 >= 0 (n ≥ 0); < 0|ψn = 0 (n ≥ 0); < 0|ψ∗

n = 0 (n < 0)

One could ask for ∆ as before with ab = a b = a ∨ b + (a|b); note(5.50)

[ψn, ψm]+ = [ψ∗m, ψ∗

n]+ = 0; [ψm, ψ∗n]+ = δmn; ψmψ∗

n =: ψmψ∗n : +(ψm|ψ∗

n);

: ψmψn∗ := ψmψ∗n− < 0|ψmψ∗

n|0 >⇒ (ψm|ψ∗n) =< 0|ψmψ∗

n|0 >

Note also ψm ∈ Wan for m < 0 so ψmψ∗n has an annihilation operator on the

left. Thus consider : ψmψ∗n := ψmψ∗

n− < 0|ψmψ∗n|0 >. For m ≥ 0 or n ≥ 0

the bracket < >= 0; otherwise m < 0 and n < 0 which means ψm ∈ Wan andψ∗

n ∈ Wcr so < 0|ψmψ∗n|0 >= [ψm, ψ∗

n]+ and : ψmψ∗n := −ψ∗

nψm. Recall we havea Clifford algebra here with [ψm, ψ∗

n]+ = δmn. This says that < 0|ψmψ∗n|0 > is

antisymmetric. Recall also H(t) =∑∞

1 t∑

n∈Z ψnψ∗n+ =

∑t∑

n∈Z : ψnψ∗n+ :.

Also H(t)|0 >= 0 and < 0|H(t) = 0; to see this note for n > 0 one has ψ∗n+|0 >= 0

while for n < 0 one has : ψnψ∗n+ := −ψ∗

n+ψn and ψn|0 >= 0. For < 0|H(t)consider for n > 0, < 0|ψnψ∗

n+ = − < 0|ψ∗n+ψn = 0 when n < 0 and n + ≥ 0.

Note also < 0|H−n = 0 for n > 0 since H−n =∑

Z ψjψ∗j−n and < 0|ψjψ

∗j−n =

− < 0|ψ∗j−nψj = 0 for j < 0. Recall also

(5.51) [∑

aij : ψiψ∗j :,

∑bij : ψiψ

∗j :] =

∑cij : ψiψ

∗j : +c0;

cij =∑

k

aikbkj −∑

k

bikakj ; c0 =∑

i<0,j≥0

aijbji −∑

i≥0,j<0

aijbji

One could ask now about the meaning in QFT (if any) of e.g.

ψ(z) =∑

ψnzn; ψ∗(z); ψ(t, z) = (X(z)τ(t)/τ(t); τ(t) =< 0|exp(H(t)g|0 >;

exp(H(t))ψ(z)exp(−H(t)) = exp(ξ(t, z))ψ(z);

< m|exp(H(t))ψ(z) = zm−1X(z) < m− 1|exp(H(t))

(cf.

of H(t) since τ(t, v) =< 0|exp(H(t))g|0 > with v = g|0 > (H(t) ∼

(and QFT) following work of Brouder, et al, Fauser et al, and Effros, et al shouldbe worth exploring

QFT and we mention one possible direction. Thus

Connections of tau functions to Hopf and Clifford algebras


some further input is needed if we are toproperly express some putative “role” of tau functions in QFT (cf. [191, 211,662]). The most obvious role seems to be in terms of the partition function ofa matrix model for example. It seems that in order to gain any true insight intothe tau function we need more perspective and one could begin by specifying thevarious known ways in which the tau function arises (beyond its emergence in thesoliton mathematics of the Japanese school (Date, Hirota, Jimbo, Miwa, Sato,et. al. and more recently Shiota, Takasaki, Takebe, et. al.). We will refer to[132, 142, 147, 191, 200, 205, 211, 564, 565, 566, 568, 569, 635, 652,653, 996, 1009] for tau functions and complex analysis, [925] for general ideasinvolving Riemann surfaces, [20, 21, 137, 142, 340, 455, 456, 542, 569, 737,738, 739, 1017] for matrix models, [567, 600] for tau functions, Hurwitz spaces,and Frobenius manifolds, and [638] for twistor connections.

REMARK 8.5.5. We want to mention here the beautiful paper [335]. Thisgives an approach to q-Feynman diagrms using combinatorics and graph theory;it delves into q-Fock spaces, q-Wick formulas, q-Gaussians, etc. and is thoroughlylovely. Moreover the analysis of partitions, graphs, and symmetric functions leadsone to the area of free probability theory and related topics for which we refer to[45, 114, 146, 391, 406, 433, 818, 904, 908, 966]. We wrote up a sketch of[335] with additional notes on free probability theory but it is too long to inserthere.

6. VERTEX OPERATORS AND SYMMETRIC FUNCTIONS

For references here see [92, 513, 516, 546] and we begin wih [516]. Oneconstructs first a family of symmetric functions generalizing the Hall-Littlewoodsymmetric functions and derives a raising operator formula. Thus let ann∈N

be a set of indeterminates and H an infinite dimensional Heisenberg algebra withinfinitely many parameters ai over C generated by hn, n ∈ Z× = Z/0 and acentral element c such that () [hm, hn] = (m/a|m|)δm,−nc. The vector spaceV is the polynomial algebra generated by the h−n (n ∈ N) and H acts on V bythe basic representation where h−n ∼ multiplication operator and hn acts as adifferentiation operator subject to the commutation relations (). One definesvertex operators on V [z, z−1] via

(6.1) X(z) = exp

( ∞∑1

an

nh−nzn

)exp

(−

∞∑1

an

nhnz−n

)=∑n∈Z

Xnz−n;

X∗(z) = exp

(−

∞∑1

an

nh−nzn

)exp

( ∞∑1

an

nhnz−n

)=∑n∈Z

X∗−nzn

The space V is a graded algebra with respect to the grading

(6.2) V = ⊕n∈ZVn, Vn = v ∈ V ; deg(v) = n, deg(hi1−1 · · ·hik

−k) = i1 + · · ·+kik

The operators Xn, X∗n act on the space via Xn : Vm → Vm+n, X∗

n : Vm →Vm−n. The normal product : : is defined as moving the elements hn to the right

It seems clear however that

6. VERTEX OPERATORS AND SYMMETRIC FUNCTIONS 359

of the elements h−n and in order to calculate the product of vertex operators oneintroduces the analytic function

(6.3) f(x) = exp

(−

∞∑1

an

nxn

)=

∞∏1

exp(−an

nxn)

As long as the variable |x| < limn→∞|an/an+1| the infinite product convergesabsolutely. Thus f(x) will be analytic in the corresponding region. There aretwo cases of special interest when an = qn and an = (1− tn)/(1− qn) (assumingt, q > 0). The convergence radius is q−1 in the first case and for the second case

• 1 for t < 1, q < 1• q/t for t > 1, q > 1• 1/t for t > 1, q < 1• 1/q for t < 1, q > 1

One writes now λ ( n to represent a partition of n of the form λ = (λ1, λ2, · · · )with n = λ1 + λ2 + · · · . The q numbers and factorials follow the standard rules

(6.4) (a; q)n = (1− a)(1− aq) · · · (1− aqn−1); (a; q)∞ =∏n≥0

(1− aqn)

The space V can be identified with the ring of symmetric functions in infinitelymany variables xi by mapping hn to the nth power sum symmetric function pn =∑

xni . The ring of symmetric functions has various useful bases, e.g. the Q

basis of power sum symmetric functions pλ = pλ1 · · · pλk; the Z basis of monomial

symmetric functions mλ =∑

xλ1i1· · ·xλk

ik; The Z basis of complete symmetric

functions cλ = cλ1 · · · cλkwith cr =

∑|λ|=r mλ; or the Z basis of Schur functions

sλ = det(cλi−i+j = mλ + · · · which can also be determined via

(6.5) sλ(x1, · · · , xn) =

∑w∈Sn

sgn(w)xw(λ+δ)∏i<j(xi − xj)

(δ = (n − 1, n − 2, · · · , 1, 0)). Using Schur functions one can give an explicitexpression for f(x) and f(x)−1. Thus

(6.6) f(x) =∑n≥0

fnxn =∑n≥0

(∑λn

(−1)(λ) aλ1 · · · aλk

zλ

)xn

where

(6.7) zλ(a) =∏i≥1

imimi!(aλ1 · · · aλk)−1 (λ ∼ (1m12m2 · · · ))

Here the fn are Schur functions s1n in terms of power sum variables an, e.g.

(6.8) f1 = −a1; f2 = −a2

2+

a21

2; f3 = −a2

2+

a1a2

2− a3

1

6; · · ·

Similarly f(x)−1 =∑

n≥0 gnxn with gn = sn, i.e.

(6.9) gn =∑λn

aλ1 · · · aλk

zλ

We will also need the symmetric function Qn defined as the vector X−n · 1 in V,i.e.

(6.10)∞∑0

Qnzn = exp

( ∞∑1

an

nh−nzn

)which is a generalized homogeneous symmetric function. One also writes qλ =Qλ1 · · ·Qλk

and notes that X(z)X(w) =: X(z)X(w) : f(w/z) where normal or-dering moves the hn to the right of the h−n (roughly hn ∼ ∂/∂h−n).

One proves next that for any tuple µ = (µ1, µ2, · · · , µk)

(6.11) X−µ = X−µ1 · · ·X−µk· 1 =

∏i<j

f(Rij)Qµ1 · · ·Qµk;

Rij(Qµ1 · · ·Qµk) = Qµ1 · · ·Qµi+1 · · ·Qµj−1 · · ·Qµk

REMARK 8.6.1. Here and in other places the first paper in [516] is sketchyand there are typos. However it does survey a lot of material so we continue evenif some confusion arises. To review the whole subject of vertex operators andsymmetric functions would be an enormous task and we only want to select a fewareas for delectation under the possible illusion that upon typing this out I willlearn what is going on.

The proof of (6.11) involves the assertion that X−λ is the coefficient of zλ =zλ11 · · · zλk

k in the expression

(6.12)∏i<j

f(zj/zi) : X(z1) · · ·X(zk) :

where |z1| > ρ−1z2 > · · · > ρ−k+1|zk| and ρ is the radius of convergence. Inmost cases one can assume ρ = 1 where the symmetric function X−λ can beexpressed as a contour integral with the above function (6.12) as the integrandand the integration contours are concentric circles with radius |z1| > · · · > |zk|.The symmetric functions X−λ form a basis in V and with various choices of thean they are or are closely related to some of the well known orthogonal symmetricfunctions.

• If an = 1 then f(x) = 1 − x and X−λ ∼ Schur symmetric functionsassociated to the partitions λ

• If an = 1− tn then f(x) = (1− x)/(1− tx) and X−λ ∼ Hall-Littlewoodsymmetric functions for the partitions λ

• If an = (1 − tn)/(1 − qn) then f(x) = (x; q)∞/(tx; q)∞ ∼ Macdonaldsymmetric functions via basic hypergeometric functions (at least for tworow partitions). This case contains all the above cases and especially theJack polynomials corresponding to an = α.

• Below one treats also an = qn which originates from the vertex represen-tations of quantum affine algebras

Now define a Hermitian structure in v via h∗n = h−n so that the power sum

symmetric functions h−λ = h−λ1 · · ·h−λkare orthogonal in the sense that <

h−λ, h−µ >= δλµzλ(a) (see above). The generalized orthogonal symmetric func-tions are defined by

(6.13) Pλ(a, x) = mλ(x) +∑λ>′µ

cλµmµ(x)

where >′ is a fixed total order (e.g. inverse lexicographic order) compatible to thedominance order on the partitions and mλ = xλ1

1 · · ·xλk

k + · · · are the monomialsymmetric functions (cf. [613]). The dominance order > is defined by comparingtwo partitions via their partial sums of the respective parts; the compatibilitymeans that if λ ′µ then λ > µ. This definition seems to depend on the total orderbut Macdonald polynomials have the triangularity relation for dominance orderand thus they are unique up to any total (or partial) order compatible to thedominance order (somewhat mysterious). We are actually interested in the dualbasis Qλ i.e. < Pλ, Qµ >= δλµ which can be rephrased via(6.14)∏

i,j

f(xiyj) =∑

λ

1zλ(a)

hλ(x)hλ(y) =∑

Pλ(a, x)Qλ(a, y) =∑

λ

mλ(x)qλ(y)

One notes that the X−λ are not orthogonal in general although it is the case forthe one row partition (i.e. X−n ·1 = Qn). In fact the inner product < X−λ, X−µ >is the coefficient of zλwµ in the series expansion

(6.15)∏i<j

f

(zi

zj

)∏i<j

f

(wi

wj

)∏i,j

f(ziwj)−1

To see this one notes that from the product of vertex operators follows that <X−λ · 1, X−µ · 1 > is the coefficient of zλwµ in the inner product

(6.16) < X(z1) · · ·X(zk) · 1, X(w1) · · ·X(w) · 1 >=

=<: X(z1) · · ·X(zk) : 1, : X(w1, · · ·X(w) : 1 > ×

×∏ij

f

(zj

zi

)f

(wj

wi

)=∏i<j

f

(zi

zj

)∏i<j

f

(wi

wj

)∏ij

f(ziwj)−1

When an = 1 − tn the function f(x) = (1 − x)/(1 − tx) and one has provedelsewhere that the matrix coefficients < X−λ, X−µ > are zero except when λ = µwhich means that the X−λ are the Hall-Littlewood functions. In general onecan calculate general matrix coefficients as follows. For an element h−λ in V weassociate a sequence of nonnegative integers m = m(λ) = (m1,m2, · · · ) where mi

is the multiplicity of i in the partition and write |m >= h−λ = hm1−1hm2

−2 · · · . Let< m| = hλ be the dual basis and then for any sequences r and s one has

(6.17) < r|X(z)|s >=∞∏1

znrn(−z−1)nsn2F0

(−rn,−sn;−;− n

an

)

where 2F0

(−n,−x;−;− 1

a

)= C

(a)n (x) is a Charlier polynomial. More generally

for real αi one has(6.18)

< r|Xα1(z1) · · ·Xαk(zk)|s >=

∏i,j

f

(aj

ai

)αiαj ∞∏1

(∑i

αizn

)rn(−∑

i

αiz−n

)sn

×

×2F0

⎛⎝−rn,−sn;−;−n

⎛⎝∑i,j

αiαjzn

i

znj

⎞⎠ /an

⎞⎠As for classical symmetric note that using < h−λ, h−ν >= δλµzλ(a) one can

realize Schur and Hall-Littlewood functions in V. For practice take the case ofMacdonald polynomials as an illustration. Thus introduce vertex operators

(6.19) s(z) = exp

( ∞∑1

1n

h−nzn

)exp

(−

∞∑1

an

nhnz−n

)=∑n∈Z

snz−n;

H(z) = exp

( ∞∑1

1− tn

nh−nzn

)exp

(−

∞∑1

an

nz−n

)=∑n∈Z

H−nzn;

S(z) = exp

(∑ 1− tn

nh−nzn

)exp

(−

∞∑1

an

n(1− tn)hnz−n

)=∑n∈Z

Snz−n

which realize the HL functions in variables xi and the Schur functions in fictitiousvariables ξi via

∏i(1 − ξiy) =

∏i(1 − txiy)/(1 − xiy). In order to find vertex

operators giving the dual symmetric functions one looks at dual vertex operatorsdefined via

(6.20) s(z) = exp

⎛⎝∑n≥1

an

nh−nzn

⎞⎠ exp

⎛⎝−∑n≥1

1n

hnz−n

⎞⎠ ;

H(z) = exp

⎛⎝∑n≥1

an

nh−nzn

⎞⎠ exp

⎛⎝−∑n≥1

1− tn

nhnz−n

⎞⎠ ;

S(z) = exp

⎛⎝∑n≥1

an

(1− tn)nh−nzn

⎞⎠ exp

⎛⎝−∑1

1− tn

nhnz−n

⎞⎠Moreover s−µ is the Schur function associated to the function

∏i f(xi)−1. To

prove that these are dual operators for s, H, S one considers

(6.21) Y (z) = exp

( ∞∑1

an

nbnh−nzn

)exp

(−

∞∑1

1− pn

nbnhnz−n

);

Y (z) = exp

( ∞∑1

1− pn

nbnh−nzn

)exp

(−

∞∑1

an

nbnhnz−n

)

with bn, p two independent scalars. One computes easily

(6.22) Y (z)Y (w) =: Y (z)Y (w) :z − w

z − pw; Y (z)Y (w) =

=: Y (z)Y (w) :z − pw

z − w; Y ∗

h (z) = Yh(z−1)

The contraction functions are exactly those for HL functions and it follows thenfrom the orthogonality of the HL functions that < Y−λ, Y−µ >= 0 except whenλ = µ so Y and Y are dual operators. The specific results (6.19) follow by taking(6.23)

(1) bn = a−1n , p = 0; (2) bn = (1− tn)/an, p = t; (3) bn = (1− tn)/an p = 0

REMARK 8.6.2. Other meaningful orthogonal operators may be obtainingby specializing bn and p. For example consider (A) bn = a−1

n , (B) bn = 1 −qn, (C) bn = 1/an(1− qn), (4) bn = 1. For case (A) one has e.g.

(6.24) Y (z) = exp

⎛⎝∑n≥1

1n

h−nzn

⎞⎠ exp

⎛⎝−∑n≥1

(1− pn)an

nhnz−n

⎞⎠There are typos here in [516].

Going back to the usual Heisenberg algebra with [hm, hn] = mδm,−nc one takesV as the symmetric algebra generated by the elements h−n (n ∈ N). Howeverthere is is difference now in that one is changing the underlying inner product bytaking Schur functions as the basic orthogonal symmetric functions instead of thegeneralized HL polynomials (usually Macdonald functions). The vertex operatorapproach is more suitable now in that some of the fundamental results beocmeextremely simple. Define

(6.25) s(z) = exp

( ∞∑1

1n

h−nzn

)exp

(−

∞∑1

1n

hnz−n

);

H(z) = exp

( ∞∑1

1− tn

nh−nzn

)exp

(−

∞∑1

1n

hnz−n

);

Q(z) = exp

( ∞∑1

1− tn

n(1− qn)h−nzn

)exp

(−

∞∑1

1n

hnz−n

)These are the vertex operators associated to the Schur functions, HL functions,and Macdonald functions respectively. Here the difference is that the elements hn

satisfy the new commutation relations above. More generally let

(6.26) X(z) = exp

( ∞∑1

an

nh−nzn

)exp

(−

∞∑1

1n

hnz−n

);

X∗(z) = exp

(−

∞∑1

1n

h−nzn

)exp

( ∞∑1

an

nhnz−n

)

One has again (6.7) and for any tuple of integers µ = (µ1, · · · , µk)(6.27)

X−µ = X−µ1 · · ·X−µk=∏i<j

F (Rij)qµ; X∗−µ = X∗

−µ1· · ·X−µk

=∏i<j

F (Rij)eµ

where qλ = Qλ1 · · ·Qλkare defined in (6.10) and en is the elementary symmetric

function. It follows that the vertex operator s(z) is a self dual operator underh∗

n = h−n and the dual of H(z) is(6.28)

H(z) = exp

( ∞∑1

1n

h−nzn

)exp

(−

∞∑1

1− tn

nhnz−n

); < H−λ, H−µ >= δλµbλ(t)

where bλ(t) =∏

i≥1(t; t)mi(λ). Consequently < sλ, H−µ > are the Kostka-Foulkespolynomials mλ,µ.

The vertex operator representations of the quantum affine algebras are con-nected to the case an = qn. To describe this we use again [hm, hn] = mδm,−nc (m,n ∈Z) and define(6.29)

X±(z) = exp

(±

∞∑1

q∓n

nh−nzn

)exp

(∓

∞∑1

q∓n

nhnz−n

)=

∑n∈Z+

X±n z−n

The commutation relations are

(6.30) X±(z)X∓(w) =: X±(z)X±(w) : [1− (w/z)]−1; X±(z)X±(w) =

=: X±(z)X±(w) : [1− (q∓(w/z)]; [X±(z), X∓(w)] =: X±(z)X∓(w) : δ(w/z)where as usual δ(x) =

∑n∈Z xn. It follows that the symmetric function X±

−λ isgiven by

(6.31) X±−λ = X±

−λ1· · ·X±

−λk· 1 =

∏i<j

(1− q∓Rij)Qλ1 · · ·Qλkq∓|λ|/2

where the Qn is the complete symmetric function (cn) or the Qn defined withan = 1. Note that when q = 1 we obtain the Schur function so this is a deforma-tion around q = 1 while the HL function is a deformation of the Schur functionaround t = 0. There are many other formulas which we omit. There are no earthshaking conclusions here; we have simply exhibited a number of formulas involvingcombinatorics, probability, QFT, tau functions, symmetric functions, vertex oper-ators, etc. We believe that these will all be important in developing the programsuggested in [662].

APPENDIX A

DeDONDER-WEYL THEORY

We extract here from [586] where a lovely discussion of Lagrangian systemscan be found. Roughly let L(x, z, v) is the Lagrangian density for a system withx = (x1, · · · , xm), z = (z1, · · · , zn), with za = fa(x) on the extremals and v =(v1

1 · · · , vnm) where va

µ = ∂µfa(x). Set πµa = ∂L/∂va

µ and H = πµava

µ−L. Then thedeDonder-Weyl Hamilton-Jacobi (DWHJ) equation is

(A.1) ∂µSµ(x, z) + H(x, z, πµa − ∂aSµ) = 0; πµ

a = ∂aSµ(x, z) = ∂Sµ/∂za

Solutions Sµ(x, z), zb = f b(x) lead to conserved currents Gµ = ∂Sµ/∂a(x, z =f(x)) (a = 1, · · · , n) In [586] one reformulates matters a la LePage [599] and wesketch only a few aspects here (standard differential geometry is mainly assumed tobe known). One takes M to be an -dimensional differential manifold with tangentvectors Y (t) and local coordinates y so Y ∼ aλ(y)∂λ with Yp ∼ Yy ∈ Ty(p). Integralcurves of Y are given via dyλ/dt = aλ(y) with yλ(0) = yλ

0 (λ = 1, · · · , ). For M2n

with coordinates y = (q1, · · · , qn, p1, · · · , pn) and Hamiltonian function H(q, p) theintegral curves φt of the vector field YH = (∂H/∂pi)(∂/∂qi) − (∂H/∂qi)(∂/∂pi)define Hamiltonian flows. Each 1-parameter local transformation group φt : y →φt(y) induces a vector field on M via

(A.2) Y f(y) = limt→01t[f(φt(y))− f(y)]

One forms (or Pfaffian forms) ω = bλ(y)dyλ are dual to vector fields and one writesω(Y ) = bλ(y)aλ(y). The wedge spaces ∧nT ∗(M ) ∼ ∧n are defined as usual andwe recall

(A.3) ω1 ∧ ω2 = (−1)pqω2 ∧ ω1; (ω1 ∈ ∧1, ω2 ∈ ∧q); dim∧p =(

p

)Given a map φ : M → Nk one has maps φ∗ : T (M ) → T (Nk) and φ∗ :T ∗(Nk) → T ∗(M ) satisfying φ∗ω(Y1, · · · , Yp) = ω(φ∗(Y1), · · · , φ∗(Yp)). Furtherd : ∧p → ∧p+1 satisfies d(ω1 ∧ ω2) = dω1 ∧ ω2 + (−1)pω1 ∧ ω2 for ω1 ∈ ∧p whiled(dω) = 0 (where df = ∂λfdyλ). In addition dω(Y1, Y2) = Y1ω(Y2) − Y2ω(Y1) −ω([Y1, Y2]) for ω ∈ ∧1. Then dφ∗(ω) = φ∗(dω) and (Poincare lemma) if dω = 0in a contractible region (ω ∈ ∧p) then there exists θ ∈ ∧p−1 such that ω = dθ.Interior multiplication is defined via i(Y ) : ∧p → ∧p−1 where (ω1 ∈ ∧p)

(A.4) i(Y )f = 0; i(Y )df = Y f ; i(Y )(ω1 ∧ ω2) = (i(Y )ω1) ∧ ω2+

+(−1)pω1 ∧ (i(Y )ω2); (i(Y )ω)(Y1, · · · , Yp−1) = ω(Y, Y1, · · · , Yp−1)

365

366 APPENDIX A

The Lie derivative is defined via

(A.5) L(Y ) = i(Y )d + di(Y ); L(Y )f = Y f ; L(Y )df = d(Y f);

L(Y )ω = limt→01t(φ∗

t ω − ω)

where φt is generated by Y. One recalls also∫

φ(G)=∫

Gφ∗ω and (Stokes’ theorem)∫

Gdω =

∫∂G

ω for suitable G. In particular if φτ (y) is a 1-parameter “variation”on M with φτ=0(y) = y and V is the vector field generated by this variation then

(A.6) Aτ =∫

φτ (G)

=∫

G

φ∗τω;

dAτ

dτ= lim

t→0

1τ

(Aτ −A0) =

=∫

limt→0(φ∗τω − ω) =

∫G

L(V )ω =∫

G

i(V )dω +∫

∂G

i(V )ω

One can treat many differential problems on manifolds via differential forms (Pfaf-fian systems - cf. [214, 526]). Thus linearly independent vector fields X1, · · · , Xm

on M define a system of partial differential equations (PDE) whose solutionsform an m-dimensional integral submanifold Im(X1, · · · , Xm) if certain integra-bility conditions hold, namely [Xµ, Xν ] = 0 (Frobenius integrability criterion).This can be restated in terms of differential forms as follows. Assume e.g. Xµ =∂µ + φa

µ(x, z)∂a (µ = 1, · · · ,m) where x = (x1, · · · , xm) and z = (z1, · · · , zn) withm+n = . One wants 1-forms ωa (a = 1, · · · , n) such that ωa(Xµ) = 0 (satisfied ifωa = dza−φa

µdxµ). The forms ωa generate an ideal I[ωa] in ∧ = ∧0⊕∧1⊕· · ·⊕∧

(i.e. if ωa vanishes on a set of vector fields Xµ so does ω∧ωa for any ω ∈ ∧). Theintegrability condition for the Xµ is equivalent to dω ∈ I[ωa] if ω ∈ I[ωa]. Nextif ω ∈ ∧p the minimal number r of linearly independent 1-forms θρ = fρ

λ(y)dyλ

by means of which ω can be expressed is called the rank of ω. If r = p then ω issimple, i.e. ω = θ1 ∧ · · · ∧ θp. At each point y the θρ generate an r-dimensionalsubspace of T ∗

y and the union of these subspaces generated by the θρ is calledA∗(ω). This determines an −r dimensional system A(ω) ⊂ T (M ) of vector fieldsY = aλ(y)∂λ satisfying θρ(Y ) = fρ

λaλ = 0. Thus Y is associated with ω if and onlyif i(Y )ω = 0. Note here that A∗(ω) is generated by the 1-forms i(∂λp−1) · · · i(∂λ1)ωwhere λ1, · · · , λp−1 = 1, · · · , l. Indeed, since i(Y1)i(Y2) = −i(Y2)i(Y1) one has

i(Y )[i(∂λp−1) · · · i(∂λ1)ω] = (−1)p−1i(∂λp−1) · · · i(∂λ1)i(Y )ω = 0

A completely integrable differential system C(ω) = A(ω) ∩ A(dω) is associatedwith ω and C∗(ω) = A∗(ω) ∪ A∗(dω) is the space of 1-forms annihilating C(ω)(characteristic Pfaffian system of ω). Evidently Y ∈ C(ω) ⇐⇒ i(Y )ω = 0 =i(Y )dω which is equivalent to i(Y )ω = 0 = L(Y )ω.

EXAMPLE A.1. Consider the form θ = −Hdt + pjdqj on R2n+1 with

(A.7) dθ = −dH(t, q, p) ∧ dt + ddp− j ∧ dqj = −(∂jHdqj+

+∂H

∂pjdpj + ∂tHdt) ∧ dt + dpj ∧ dqj = (dpj + ∂jHdt) ∧

(dqj − ∂H

∂pjdt

)

D DONDER-WEYL THEORY 367

Its associated system (= characteristic system since dθ is exact) is generated bythe 2n Pfaffian forms

(A.8) i(∂j)dθ = −(dpj + ∂jHdt) = −θj ; i(∂/∂pj) = dqj − ∂H

∂pjdt = ωj

If there are no additional constraints the rank and class of dθ are 2n and thespace A(dθ) = C(dθ) of (characteristic) vector fields associated with dθ is 1-dimensional and can be generated via XH = ∂j + (∂H/∂pj)∂j − (∂H/∂qj)(∂/∂pj)(since θj(XH) = 0 = ωj(XH) = 0. Thus the associated integral manifolds of dθare the 1-dimensional solutions qj = ∂H/∂pj and pj = −∂H/∂qj .

We recall next the Legendre transformation vj → pj with L → H. Thusthe time evolution of a system involves a curve C0(t) based on q = dq/dt in say[t1, t2]×Gn = Gn+1. For suitable L the curve is characterized by making the ac-tion integral A[C] =

∫ t2t1

L(t, q, q)dt stationary, leading to the Euler-Lagrange (EL)equations (d/dt)(∂L/∂qj) − ∂jL = 0 for (j = 1, · · · , n). This can be rephrasedas follows (with a view toward the Lepage theory). Consider the LagrangianL = L(t, q, v) − λj(vj − qj) with EL equations ∂L/∂vj = ∂L/∂vj − λj = 0(since ∂L/∂vj = 0) leading to L(t, q, q, v) = L(t, q, v) − vj∂L/∂vj + qj∂L/∂vj .Then introduce new variables pj = ∂L/∂vj and assume the matrix (∂pj/∂vk) =(∂2L/∂vk∂vj) to be regular. Solve the equations pj = ∂L/∂vj for vj = φ(t, q, p)and define H(t, q, p) = φ(t, q, p)pj −L[t, q, v = φ(t, q, p)]. Then one obtains for theLagrangian L the result L(t, q, q, p) = −H(t, q, p)+ qjpj which contains 2n depen-dent variables q, p but depends only on the derivatives in q (not in p). Hence the ELequations are −∂L/∂pj = ∂H/∂pj − qj = 0 and (d/dt)(∂L/∂dotqj) − ∂L/∂qj) =pj + ∂jH = 0.

The Lepage method is now similar in spirit to this introduction of Hamil-ton’s function H(t, q, p) and the resulting derivation of the canonical equationsof motion, but via the use of differential forms, it is more efficient. Thus letω = L(t, q, q)dt and the introduction of new variables vj equal to qj on the ex-tremals is equivalent to introducing 1-forms ωj = dqj − vjdt which vanish on theextremals q(t) where dqj(t) = qjdt, or where the tangent vectors et = ∂t + qj∂j areannihilated by each ωj (i.e. ωj(et) = qj−vj = 0). Thus as far as the extremals areconcerned the form ω = Ldt is only one representative in an equivalence class of1-forms, the most general of which is Ω = L(t, q, v)dt + hjω

j (note the n Pfaffianforms ωj generate an ideal I[ωj ] which vanishes on the extremals). The coeffi-cients hj (which correspond to Lagrange multipliers λj above) can be arbitraryfunctions of t, q, v and according to the Lepage theory they can be determined bythe following condition. Thus note

(A.9) dΩ =(

∂L

∂vj− hj

)dvj ∧ dt + (−∂jLdt + dhj) ∧ ωj =

=(

∂L

∂vj− hj

)dvj ∧ dt (mod I[ωj ])

e

368 APPENDIX A

We therefore have dΩ = 0 on the extremals where ωj = 0 if and only if hj =∂L/∂vj = pj . Inserting this value for hj into Ω one obtains

(A.10) Ω = Ldt + pjωj = Ldt + pj(dqj − vjdt) = −Hdt + pjdqj = θ

These equations provide an interpretation of the Legendre transformation via vj →pj = (∂L/∂vj)(t, q, v) with L → H = vjpj − L (this can be implemented by achange of basis dt → dt and ωj → dqj) and this point of view is of importance infield theories.

Now one goes to the implications of generalizing Lepage’s equivalence relationθ ≡ Ldt (mod I[ωj ]) and dθ ≡ 0 (mod I[ωj ]) for θ = Ldt + pjω

j = −Hdt + pjdqj

to a field theory context. Thus working with 2 independent variables for simplicitylet xµ µ = 1, 2 be independent variables and za, a = 1, · · · , n those variables ofy = (x1, x2, z1, · · · , zn) which become the dependent variables za = fa(x) on 2-dimensional submanifolds Σ2. Further the variables va

µ become the derivatives∂µza(x) = ∂µfa(x) on Σ2 and especially on the extremals Σ2

0 defined below. Thislast property says that the forms ωa = dza − va

µdxµ vanish on the extremals Σ20

where vaµ = ∂µza(x). Consider the action integral A[Σ2] =

∫G2 L(x, z, ∂z)dx1dx2

which is supposed to become stationary for the extremals Σ20 when we vary the

functions za(x) and ∂µza(x) (see below). By using the forms ωa = dza−vaµdxµ one

can argue here as before with mechanics. As far as extremals go the Lagrangian2-form ω = L(x, z, v)dx1 ∧ dx2 is one representative in an equivalence class of2-forms which can be written as

(A.11) Ω = ω + h1aωa ∧ dx2 + h2

adxx1 ∧ ωa +12habω

a ∧ ωb ≡

ω (mod I[ωa]); hba = −hab

The term with hab is new compared to mechanics and is only possible because Ωis a 2-form and if n ≥ 2. In mechanics the coefficients hj in Ω = Ldt + hjω

j weredetermined by the requirement dΩ = 0 mod I[ωj ] and for field theories we have

(A.12) dΩ =(

∂aLdza +∂L

∂vaµ

dvaµ

)∧ dx1 ∧ dx2 + dh1

a ∧ ωa ∧ dx2−

−h1adva

1 ∧ dx1 ∧ dx2 + dh2a ∧ dx2 ∧ ωa + dh2

a ∧ dx1 ∧ ωa + h2adx1 ∧ dva

2 ∧ dx2+

+12d(habω

a ∧ ωb) =(

∂

∂vaµ

− hµa

)dva

µ ∧ dx1 ∧ dx2 + O(mod I[ωa])

where the equality dza ∧ dx1 ∧ dx2 = ωa ∧ dx1 ∧ dx2 has been used. Thus thecondition dΩ = 0 mod I[ωa] is equivalent to hµ

a = ∂L/∂vaµ = πµ

a (as in mechanics).What is new however is that the condition dΩ = 0 mod I[ωa] does not impose anyrestrictions on hab and this has far reaching consequences. Indeed inserting theformula for hµ

a = πaµ into (A.11) one gets

(A.13) Ω = Ldx1 ∧ dx2 + π1aωa ∧ dx2 + π2

adx1 ∧ ωa + (1/2)habωa ∧ ωb

Each choice of hab defines a canonical form Ωh and the implications will be seen.First introduce some notational simplifications via aµ = Ldxµ + πµ

aωa and εµν =

369

εµν , ε12 = 1, and εµν = −ενµ. Then (A.13) can be written as

(A.14) Ω = Ldx1 ∧ dx2 + πµaωa ∧ dΣµ + (1/2)habω

a ∧ ωb =

= aµ ∧ dΣµ − Ldx1 ∧ dx2 + (1/2)habωa ∧ ωb; dΣµ = εµνdxν

Now recall that the Legendre transformation vj → pj , L → H was imple-mented via a change of basis dt → dt, ωj → dqj by inserting for ωj the expressiondqj − vjdt and subsequent identifications. Generalizing this procedure to (A.14)means that we replace ωa by dza − va

µdxµ and write Ω in terms of the basisdx1 ∧ dx2, dza ∧ dx2, dx1 ∧ dza, and dza ∧ dzb to get

(A.15) Ω = [L− πµava

µ + (1/2)hab(va1vb

2 − va2vb

1)dx1 ∧ dx2+

+(π1a − habv

b2)dza ∧ dx2 + (π2

a + habvb1)dx1 ∧ dza + (1/2)habdza ∧ dzb

Define now hµνab = εµνhab and rewrite the last equation as

(A.16) Ω = (L− πµava

µ + (1/2)hµνab va

µvbν)dx1 ∧ dx2+

+(πµa − hµν

ab vbν)dza ∧ dΣµ + (1/2)habdza ∧ dzb

The generalized Legendre transformation (a la Lepage) defines the canonical mo-menta pµ

a as pµa = πµ

a−hµνab vb

ν and the Hamiltonian as H = πµava

µ−(1/2)hµνab va

µvbν−L

leading to

(A.17) Ω = −Hdx1 ∧ dx2 + pµadza ∧ dΣµ + (1/2)habdza ∧ dzb

In order to express H and hab as functions of pµa the Legendre transformation

vaµ → pµ

a has to be regular, namely

(A.18)∣∣∣∣∂pµ

a

∂vbν

∣∣∣∣ =∣∣∣∣( ∂2L

∂vbν∂va

µ

− ∂hµλac

∂vbν

vcλ − hµν

ab

)∣∣∣∣ = 0

The coefficients hµνab have been arbitrary up to now and an appropriate choice

can always guarantee the inequality (A.18) which is an interesting possibility fordefining a regular Legendre transformation if the conventional one va

µ → πµa is

singular as in gauge theories.

Assume now one can solve pµa = πµ

a − hµνab vb

ν for the vaµ, i.e. va

µ = φaµ(x, z, p)

and putting this into hµνa (x, z, v) one obtains ηµν

ab (x, z, p) = hµνab (x, z, v = φ(x, z, p))

leading to

(A.19) H(x, z, p) = πµa [x, z, v = φ(x, z, p)]φa

µ(x, z, p)−

−(1/2)ηµνab φa

µφbν − L(x, z, v = φ(x, z, p)]

This leads to

(A.20) dH = vaµdπµ

a + πµadva

µ − (1/2)d(hµνab va

µvbν)− dL;

dπµa = dpµ

a + d(hµνab vb

ν)Combining these equations yields

(A.21) dH = vaµdpµ

a + πµadva

µ + (1/2)vaµvb

νdhµνab − dL

D DONDER-WEYL THEORYe

370 APPENDIX A

Inserting now into (A.21) dH = ∂µHdxµ + ∂aHdza + (∂H/∂pµa)dpµ

a where it isspecified that ∂µH = (∂H/∂xµ)|z,p fixed along with

(A.22) dL = ∂µLdxµ + ∂aLdza + πµadva

µ; dhµνab = dηµν

ab =

= ∂ληµνab dxλ + ∂cη

µνab dzc +

∂ηµνab

dpλc

dpλc

and comparing coefficients of dxµ, dza, dpµa yields then

(A.23) ∂λH − 12φa

µφbν∂ληµν

ab = −∂λL;

∂cH − 12φa

µφbν∂cη

µνab = −∂cL;

∂H

∂pλc

− 12φa

µφbν

∂ηµνab

∂pλc

= vcλ

Up to now the variables xµ, za, and vaµ or pµ

a have been treated as independentand this is no longer the case if one looks for the extremals Σ2

0, namely those2-dimensional submanifolds za = fa(x), pµ

a = gµa (x) for which the variational

derivative dA[Σ2]/dτ vanishes at τ = 0, i.e.

(A.24) A[Σ2] =∫

Σ2Ω;

dAτ

dτ

∣∣∣∣τ=0

[Σ20] =

∫Σ2

0

i(V )dΩ +∫

∂Σ20

i(V )Ω = 0

V = V µ(x)∂µ + V a

(z)∂a + V µ(p)a(∂/∂pµ

a)

Some calculations then lead to

(A.25) −∂aH − ∂µpµa − (∂µηµν

ab )∂νzb + (1/2)(∂aηµνbc )∂µzb∂νzc−

−(∂cηµνab )∂µzc∂νzb − ∂ηλν

ac

∂pµb

∂λpµb ∂νzc = 0

which are the analogue of the canonical equations pj + ∂jH = 0. If one definesnow

(A.26)d

dxµF (x, z, p) = ∂µF + ∂aF∂µza +

∂F

∂pνa

∂µpνa

(e.g. dza(x)/dxµ = ∂µza(x)) one can rewrite (A.25) as

(A.27)dpµ

a

dxµ+ ∂aH = (1/2)∂aηµν

bc ∂µzb∂νzc − (dηµνab /dxµ)∂νzb

Further there are equations

(A.28) − ∂H

∂pµa

+ ∂µza +12

∂ηλνbc

∂pµa

∂λzb∂νzc = 0

which are the same as the equations in (A.23), if ∂µza(x) = vaµ (which is a con-

sequence of (A.18)). The equations (A.27) and (A.28) represent a system ofn + 2n = 3n first order PDE for the 3n functions za = fa(x) and pµ

a = gµa (x).

By eliminating the canonical momenta pµa one obtains a system of n second order

PDE for the za, namely the EL equations (d/dxµ)(∂L/∂vaµ)− ∂aL = 0 (cf. [586]

for details and note the hab have dropped out completely). One can also showthat the inequality (A.18) implies va

µ = ∂µza(x) on the extremals (at least in aneighborhood of va

µ = 0 - recall πaµ = ∂L/∂va

µ).

REMARK A.1. One notes an important structural difference between the

371

variational system i(∂a)dΩ, i(∂/∂pµa)dΩ of differential forms which determine the

extremals here and the system i(∂j)dθ, i(∂/∂pj)dθ arising in mechanics. In thelatter case the 1-forms i(∂j)dθ, i(∂/∂pj)dθ generate the characteristic Pfaffian sys-tem C∗(dθ) of dθ; i.e. the extremals associated with the canonical form θ coincidewith the characteristic integral submanifolds of the form dθ which means that thetangent vectors ∂t + qjpj + pj(∂/∂pj) = ∂t +(∂H/∂pj)∂j − (∂H/∂qj)(∂/∂pj) forma characteristic vector field associated with dθ (i.e. this vector field is annihilatedby the forms i(∂j)dθ and i(∂/∂pj)dθ). The situation is different for field theories;here the tangent vectors Σ′

(µ) = ∂µ + ∂µza∂a + ∂µpνa(∂/∂pν

a) are annihilated in

pairs - as a 2-vector Σ′(1)∧ Σ′

(2) by the 3n+2 two forms i(∂a)dΩ, i(∂/∂pµa)dΩ, and

i(∂µ)dΩ. On the other hand the characteristic Pfaffian system C∗(dΩ) is generatedby

(A.29) i(∂µ)i(∂ν)dΩ; i(∂µ)i(∂a)dΩ; i(∂a)i(∂b)dΩ;

i(∂µ)i(∂/∂pµa)dΩ; i(∂a)i(∂/∂pν

b )dΩEach characteristic vector Y of the form dΩ (i.e. i(Y )dΩ = 0) is also annihilated bythe 2-forms i(V )dΩ where V = ∂µ, ∂a, or ∂/∂pµ

a . However if i(Y )i(V )dΩ = 0 = 0one cannot conclude that i(Y )dΩ = 0. As a simple example take n = 1 andL = T (v)− V (z) (a not unusual form in physics). Then pµ = πµ and

(A.30) Ω = Ldx1 ∧ dx2 + π1ω ∧ dx2 + π2dx1 ∧ ω =

−Hdx1 ∧ dx2 + π1dz ∧ dx2 + π2dx1 ∧ dz; ω = dz − vµdxµ; H = πµvµ − L

Then dΩ = (∂zLdx1∧dx2−dπ1∧dx2−dx1∧dπ2)∧ω is a 3-form in the 5 variablesxµ, z, vµ (µ = 1, 2). Evidently ω ∈ C∗(dΩ) but the factor

(A.31) ρ = ∂zLdx1 ∧ dx2 − dπ1 ∧ dx2 − dx1 ∧ dπ2 =

= ∂zLdx1 ∧ dx2 − ∂2L

∂v1∂vνdvν ∧ dx2 − ∂2L

∂v2∂vνdx1 ∧ dvν

is a 2-form in the 4 variables xµ and vµ with nontrivial rank 4 (cf. [586]). HencedΩ has rank 5 and the integral submanifolds of the characteristic system C∗(dΩ)are 0-dimensional, whereas the integral submanifolds (the extremals) of the vari-ational system i(∂µdΩ, i(∂a)dΩ, i(∂/∂pµ

a)dΩ are 2-dimensional in this example.Only if Ω is a 1-form will the variational system coincide with the characteristicsystem C∗(dΩ).

The essential new feature of the canonical framework above are the hab(x, z, v) =ηab(x, z, p). There are several examples: hab = 0 in which case pµ

a = πµa and H =

πµava

µ − L. If the Legendre transformation vaµ → πµ

a is regular (i.e. (∂2L/∂vaµ∂b

ν)is regular, one can express H as a function of x, z, π and the canonical equations(A.27) - (A.28) take the form ∂µπµ

a = −∂aH and ∂µza = ∂H/∂πµa = φa

µ(x, z, π).The choice hab = 0 is the conventiional one and is called the canonical theory forfields of deDonder and Weyl. For a system with just one real field this is the onlypossible one since there can be no nonvanishing hab. In more detail look at thevariational system I[i(V )dΩ0] which is generated by the 2-forms

(A.32) i(∂a)dΩ0 = −∂aHdx1 ∧ dx2 − dπµa ∧ dΣµ = λa;


372 APPENDIX A

i(∂/∂πµa )dΩ0 = ωa ∧ dΣµ = ωa

µ; i(∂µ)dΩ0 =

= ωa ∧ (∂aHdΣµ + εµρdπρa)− φa

µλa = (dh− ∂ρHdxρ) ∧ dΣµ + εµρdza ∧ dπρa = ωµ

This differential system (A.32) has some peculiar properties(1) One has

(A.33) dλa = −d(∂aH) ∧ dx1 ∧ dx2; dωaµ = −d(φa

µ) ∧ dx1 ∧ dx2;

dωµ = −d(∂µH) ∧ dx1 ∧ dx2

and since dxµ∧λa = dπµa ∧dx1∧dx2 with dxµ∧ωa

ν = −δµν dza∧dx1∧dx2

we see that dλa, dωaµ, and dωµ belong to I[λa, ωa

µ, ωµ].(2) The 3n + 2 forms λa, ωa

µ and dωµ are linearly independent.(3) If v = Aµ∂µ + Ba∂a + Cµ

a (∂/∂πµa ) is an arbitrary tangent vector then it

is a 1-dimensional integral element of the system (A.32) (which containsno 1-forms) and its “polar” system i(v)λa, i(v)ωa

µ, i(v)ωµ has at mostthe rank 2n + 1. This follows from the relations

(A.34) Aµi(v)ωaµ = (Ba −Aρφa

ρ)AµdΣµ;

Aµi(v)ωµ + Bai(v)λa + Cµa i(v)ωa

µ = i(v)i(v)dΩ0 = 0which shows that at most 2n + 2 of the 1-forms i(v)λa, i(v)ωa

µ, i(v)ωµ

are linearly independent. The maximal rank 2n+2 is relized e.g. for thevector v = ∂/∂x1.

REMARK A.2. In more conventional language the terms involving hab may beinterpreted as follows. Defining

(A.35)d

dxµ= ∂µ + va

µ∂a + vaµν

∂

∂vaν

; vaνµ = va

µν

one has, with hµνab = εµνhab, hµν

ab vaµvb

ν = hµνab d(zavb

ν)/dxµ. If in addition dhµνab /dxµ =

0 (e.g. if the hµνab are constants, then hµν

ab vaµvb

ν = d(hµνab zavb

ν) (i.e. the termhµν

ab vaµvb

ν is a total divergence. This implies that the Lagrangian L∗(x, z, v) =L(x, z, v)− (1/2)hµν

ab vaµvb

ν gives the same field equations as L itself, but the canon-ical momenta are ∂L∗/∂va

µ = πµa − hµν

ab vbν = pµ

a .

EXAMPLE A.2. A nontrivial unique choice of the hab is obtained as fol-lows. By means of 1-forms aµ = Ldxµ + πµ

aωa and θµ = −Hdxµ + pµadza one can

construct forms as in (A.13) - (A.14), namely

(A.36)Ωc = 1

La1 ∧ a2 = Ldx1 ∧ dx2 + πµaωa ∧ dΣµ + 1

2L (π1aπ2

b − π2aπ1

b )ωa ∧ ωb =

= − 1H

θ1 ∧ θ2 = −Hdx1 ∧ dx2 + pµadza ∧ dΣµ − 1

2H(p1

ap2b − p2

ap1b)dza ∧ dzb

This form Ωc is unique among all possible forms (A.13) because it is the only onewhich has the minimal rank 2 and it defines Caratheodory’s canonical theory forfields.

REMARK A.3. In mechanics if one replaces a given Lagrangian L(t, q, q) byL∗ = L + df(q)/dt = L + qj∂jf then the EL equations of L and L∗ are the same.

373

However for the canonical momenta one has p∗j = ∂L∗/∂qj = ∂L/∂qj + ∂jf =pj +∂jf so the canonical momenta are changed. Thus the canonical reformulationof n second order differential equations by a system of 2n first order differentialequations is not unique. Further in field theories the corresponding freedom is evenmore substantial if n ≥ 2. The most important classifying criterion will be therank of the form Ω for a given set of coefficients hab since for rank(Ω) = r (withdΩ = 0 on the extremals) the integral submanifolds associated with the canonicalform Ω have dimension n + 2− r (r ≥ 2). As examples consider:

• If n = 1 (i.e. only one dependent variable z) the rank of Ω is always 2.• If all hab = 0 the deDonder-Weyl canonical form

(A.37) Ω0 = aµ ∧ dΣµ − Ldx1 ∧ dx2

has rank 4 (for n ≥ 2) so the integral submanifolds of Ω0 have dimension2 + n− 4 = n− 2

• Caratheodory’s canonical form (A.36) has rank 2; it is the only canonicalform which allows for n-dimensional wave fronts.

We go now to the Hamilton-Jacobi (HJ) theory for deDonder-Weyl (DW) fields.One recalls that in mechanics the property dθ = 0 with θ = −Hdt + pjdqj onthe extremals, combined with the Poincare lemma implies the basic HJ relationdS(t, q) = −Hdt + pjdqj . In the same manner one can conclude from dΩ =0 (mod I[ωa]) that Ω is locally an exact 2-form on the extremals Σ2

0, which meansthat Ω is expressable by differentials dS1(x, z), dS2(x, z), and dx1, dx2, · · · wherethe number of these differentials equals rank ω. For example in the Caratheodorytheory where Ω = Ωc has rank 2 one has dS1 ∧ dS2 = −(1/H)θ1 ∧ θ2. For thedeDonder-Weyl theory where Ω = Ω0 has rank 4 one has

(A.38) Ω0 = dS1(x, z) ∧ dx2 + dx1 ∧ dS2(x, z) = −Hdx1 ∧ dx2 + πµadza ∧ dΣµ

which implies

(A.39) (a) πµa = ∂aSµ(x, z) = ψµ

a (x, z);

(b) ∂µSµ(x, z) = H(x, z, π = ψ(x, z))These are simple generalizations of the relations pj = ∂jS and ∂tS + H = 0 inmechanics. Further according to (A.38) the wave fronts are given by Sµ(x, z) =σµ = const. and xµ = const. because

(A.40) i(w)Ω0 = dS1(w)dx2 − w(2)dS1 + w(1)dS2 − dS2(w)dx1 = 0;

w = w(µ)∂(µ) + wa∂a; ∂(µ) = ∂/∂xµ; dSµ(w) = w(ν)∂(ν)Sµ + wa∂aSµ

implies w(µ) = 0 for µ = 1, 2 and wa∂aSµ = waπµa = 0 so the wave fronts associated

with Ω0 are (n − 2)-dimensional and lie in the characteristic planes xµ = const..Note also that (A.39)-b is a first order PDE for two functions Sµ so we can chooseone, say S2, with a large degree of arbitrariness and then solve for S1. The mainrestriction on S2 is the transversality condition that at each point (x, z = f(x))the derivatives ∂aS2(x, z) equal the canonical momenta πµ

a (x).

Another important difference between HJ theories in mechanics and in fieldtheories should be mentioned. Recall that any solution of the HJ equation in


374 APPENDIX A

mechanics leads to a system qj = φj(t, q) = (∂H/∂j)(t, q, p = ∂S(t, q)) whosesolutions qj(t) = f j(t, u) (u = (u1, · · · , un) constitute an n-parameter family ofextremals which generate S(t, q) provided |∂qj/∂uk)| = 0. Now calculate S(t, q)by computing σ(t, u) =

∫ tdτL(τ, f(τ, u), ∂τf(τ, u)), solving qj = f j(t, u) for uk =

χ(t, q), and inserting the functions χk(t, q) into σ(t, u) to get S(t, q) = σ(t, χ(t, q)).This procedure is not possible in field theories. Indeed in the DW theory one hasva

µ = ∂H/∂πµa and given any solution Sµ(x, z) of the DWHJ equation (A.39)-b we

define the slope functions

(A.41) φaµ(x, z) =

∂H

∂πµa

(x, z, πµa = ∂aSµ(x, z))

However the PDE ∂µza(x) = φaµ(x, z) will only have solutions za = fa(x) under

the integrability conditions

(A.42)d

dxνφa

µ(x, z(x)) = ∂νφaµ + ∂bφ

aµ · φb

ν =

=d

dxµφa

ν(x, z(x)) = ∂µφaν + ∂bφ

aν · φb

µ

This involves rather stringent conditions (cf. [586] for examples). Suppose how-ever that we have found solutions Sµ of the DWHJ equations (A.39)-b such thatthe slope functions (A.41) do obey the integrability conditions (A.42); then thesolutions za = fa(x) of ∂µza = φa

µ are extremals and satisfy dπµa = −∂aH as well.

To see this recall πµa (x) = ψµ

a (x, z(x)) = ∂aSµ(x, z(x)) which implies

(A.43)dπµ

a

dxµ= ∂a∂µSµ(x, z(x)) + ∂b∂aSµ(x, z(x))∂µzb(x)

On the other hand one has ∂µSµ = −H[x, z(x), π = ∂S(x, z(x)) and (dH/dza) =DaH = ∂aH + (∂H/∂πν

b )∂b∂aSν . Consequently

(A.44)dπµ

a

dxµ= −∂aH +

(∂µzb(x)− ∂H

∂πµb

)∂a∂bS

µ

which shows that the canonical equations dπµa/dxµ = −∂aH are a consequence of

(A.41).

Regarding currents note that if S(t, q, a) is a solution of the HJ equationdepending on a parameter a then G(t, q, a) = ∂S(t, q, a)/∂a is a constant of motionalong any extremal where pj(t) = ∂jS(t, q(t), a) holds. Thus (d/dt)G(, q(t), a) = 0when qj(t) = f j(t) is an extremal and to see this consider

(A.45) dS(t, q, a) = ∂tSdt + ∂jSdqj + (∂S/∂a)da =

= −H(t, q, p = ψ(t, q))dt + ψj(t, q, a)dqj + Gda; ψj = ∂jS(t, q, a)

Exterior differentiation of (A.45) gives

(A.46) 0 = −[−∂ − jHdqj +

|ppH

∂pjdψj(t, q, a)

]∧dt+dψj(t, q, a)∧dqj +dG∧da

375

Now suppose qj = qj(t) is an arbitrary curve (not necessarily an extremal); thensince dψj = ∂tψj + ∂kψjdqk + (∂ψj/∂a)da with dqj = qdt it follows from (A.46)that

(A.47)[qj − ∂H

∂pj(t, q, ψ(t, q))

]∂ψj

∂a=

d

dtG = 0

Thus if qj = ∂H/∂pj then from (A.47) follows immediately dG/dt = 0 (Noether’stheorem is a special case).

Recall that in mechanics if S(t, q, a) is a solution of the HJ equaiton dependingon n parameters aj such that

(A.48)∣∣∣∣( ∂2S

∂qj∂ak=

∂ψj

∂ak(t, q, a)

)∣∣∣∣ = 0

then one can solve the equations ∂S/∂ak = bk = const. (k = 1, · · · , n) for thecoordinates qj = f j(t, a, b) and the curves q = f(t, a, b) are extremals. This followsfrom (A.47) where G = bk = const. now and this implies (qj−(∂H/∂pj))(∂ψj/∂k) =0. In view of (A.48) the coefficients qj − (∂H/∂pj) of this homogeneous systemhave to vanish which means that the functions qj = f j(t, a, g) are solutions ofthe canonical equations qj = (∂H/∂pj)(t, q, pj = ∂jS(t, q)). It then follows fromprevious discussion that they are also solutions of pj = −∂jH as well and such asolution S of the HJ equation with the property (A.48) is called a complete inte-gral; it provides a set of solutions qj = f j(t, a, b) of the equations of motion whichdepends on the largest possible number (2n) of constants of integration (i.e. the setis complete). There is a straightforward generalization to field theories. Supposeone has a solution Sµ(x, z) of (A.39)-b depending on 2n parameters aν

b such that|(∂2Sµ/∂zc∂aν

b ) = (∂ψµc /∂aν

b )| = 0 and for which the equations (A.42) hold. Theneach of the 2n parameters aν

b will generate a current Ga;bν = (∂Sµ/∂aν

b )(x, z, a).Suppose one has 4n functions gµ;b

ν (x), arbitrary up to the two properties that(d/dxµ)gµ;b

ν = 0, such that

(A.49) Gµ;bν (x, z, a) =

∂Sµ

∂aνb

(x, z, a) = gµ;bν (x)

should be a solvable system of 4n equations for the n variables zb. Thus 3n of theequations (A.49) cannot be independent of the other n and such functions satisfy-ing (d/dxµ)gµ;b

ν (x) = 0 are not difficult to find. Indeed one can take 2n arbitrarysmooth functions hb

ν(x) with gµ;bν (x) = εµλ∂λhb

ν(x). Then one needs appropriatesolutions Sµ(x, z, a) and functions gµ;b

ν (x) such that the 4n equations (A.49) haven solutions za = fa(x); once found they will automatically be extremals since (cf.[586] for details)

(A.50) (∂µzb(x)− (∂H/∂πµb ))(∂ψµ

b /∂aνc ) = 0

Combined with the inequality above one concludes that ∂bµ(x) − (∂H/∂πµ

b ) =0 leading to fulfillment of the integrability condition (A.42) with (dπµ

a/dxµ) =−∂aH. Thus for finding solutions of the field equations via the DWHJ equation(A.39)-b one can look for solutions Sµ which fulfill the integrability condition(A.42) and then either solve the equations ∂µza = φa

µ, or, if the solution Sµ is


376 APPENDIX A

a complete integral one can try to solve the algebraic equations (A.49). Anothermethod is also indicated in [586].

For extension to m > 2 independent variables a few remarks are extractedhere from [586]. Thus take ω = Ldx1 ∧ · · · ∧ dxn as the Lagrangian m-formbelonging to an equivalence class of m-forms Ω which consist of ω plus a linearcombination of all m-forms obtained from dx1 ∧ dxm by replacing the dxµ by1-forms ωa = dza − va

µdxµ which vanish on the extremals where vaµ = ∂µfa(x)

and generate an ideal I[ωa]. More specifically consider Minkowski space withcoordinates x0, · · · , x3 (with c = 1) and forms

(A.51) Ω = Ldx0 ∧ · · · ∧ dx3 + hµz ωa ∧ d3Σµ + (1/4)hµν

ab ωa ∧ ωb ∧ d2Sµν+

+13!

habc;µωa ∧ ωb ∧ ωc ∧ dxµ +14!

habcsωa ∧ ωb ∧ ωc ∧ ωs

d3Σµ =13!

εµαβγdxα ∧ dxβ ∧ dxγ ; d2Sµν = (1/2)εµναβdxα ∧ dxβ

Here εµνρσ is totally antisymmetric with ε0123 = 1, hµνab are antisymmetric in

(µν) and (a, b) separately, habc;µ are completely antisymmetric in (a, b, c), etc.Thus the term habcs can occur only if n ≥ 4 and the h-coefficients are arbitraryfunctions of x, z, v. As before hµ

a are determined to be equal to πµa = ∂L/∂va

µ bythe requirement that dΩ = 0 (mod I[ωa]). The Legendre transformation va

µ → pµa

and L → H is again implemented by inserting on the right in (A.51) for ωa theexpression dza− va

µdxµ and identifying H with the resulting negative coefficient ofdx0 ∧ · · · ∧ dx3 while the canonical momenta pµ

a are the coefficients of dza ∧ d3Σµ.If habc;µ = 0 = habcs one obtains

(A.52) pµa = πµ

a − hµνab vb

ν ; H = πµava

µ − (1/2)hµνab va

µvbν − L

One notes the following identities (cf. [657])

(A.53) εαβγµd3Σµ = dxα ∧ dxβ ∧ dxγ ; dxρ ∧ d3Σµ = δρµdx0 ∧ · · · ∧ dx3;

dxρ ∧ d2Sµν = δρνd3Σµ − δρ

µd3Σν ; dxσ ∧ dxρ ∧ d2Sµν = (δσµδρ

ν − δσν δρ

µ)dx0 ∧ · · · dx3

If d(hµνab )dxµ = 0 where (d/dxµ) = ∂µ + va

µ∂a + vaµν∂/∂va

ν , and vaµν = va

νµ, then

(A.54) hµνab va

µvbν =

d

dxµ(hµν

ab zavbν)

For the DW canonical theory where all hµνab = 0 (A.51) becomes

(A.55)Ω0 = Ldx0 ∧ · · · ∧ dx3 + πµ

aωa ∧ d3dΣµ = aµ ∧ d3dΣµ − 3Ldx0 ∧ · · · ∧ dx3

where aµ = Ldxµ + πµaωa = −Tµ

ν dxν + πµadza. (A.55) shows that Ω0 may be ex-

pressed by the 8 linearly independent 1-forms dx0, · · · , dx3, a0, · · · , a3 and there-fore has rank 8 if n ≥ 2 (for n = 1 it has rank 4 since one p-form in p+1 variablesalways has rank p). Replacing ωa in Ω0 by dza − va

µdxµ results in

(A.56) Ω0 = −HDW dx0 ∧ · · · ∧ dx3 + πµadz0 ∧ d3Σµ; HDW = πµ

avaµ − L

377

and the DWHJ equation is obtained from

(A.57) dSµ ∧ d3Σµ = −HDW dx0 ∧ · · · ∧ dx3 + πµadza ∧ d3Σµ

which implies

(A.58) ∂µSµ(x, z) + HDW (x, z, π) = 0; πµa = ∂aSµ


APPENDIX B

RELATIVITY AND ELECTROMAGNETISM

We extract first from [12], which still appears to be the best book ever writtenon classical general relativity, and will sketch some of the essential features. Fourcriteria for field equations are stated as:

(1) Physical laws do not distinguish between accelerated systems and inertialsystems. This will hold if all laws are written in tensor form.

(2) Both gravitational forces and fictitious forces appear as Christoffel sym-bols (connection coefficients) in a mathematically similar form. This isdesirable since they should be indistinguishable in the small.

(3) The gravitational equations should be phrased in covariant tensor formand should be of second order in the components of the metric tensor.

(4) For unique solutions one wishes the field equations to be quasi-linear (i.e.the second derivatives enter linearly).

Now the signature for a Lorentz metric is taken to be (1,−1,−1,−1) and onewrites f|α = ∂f/∂xα while f||α is the covariant derivative (defined below). Weassume known here the standard techniques of differential geometry as used in[12]. Then e.g. for a contravariant (resp. covariant) vector ξi (resp. ηm) onewrites

(B.1) ξi||k = ξi

|k − Γikξ

; Γik = −

i

k

; ηm|| = ηm| +

r

m

(the bracket notation is used for Christofel symbols which are connection coeffi-cients). One defines the Riemann curvature tensor via

(B.2) Rαηβγ =

α

β η

|γ−

αη γ

|β

+

ατ γ

τ

β η

−

ατ β

τ

γ η

and a necessary (and sufficient) condition for a Riemann space to have a Lorentzmetric is Rα

ηβγ = 0 (i.e. the space is flat); this equation is in fact a field equationfor a flat and gravity free space (with Lorentz metric as a solution). Note hereξα

||β||γ − ξα||γ||β = Rα

ηβγξη which implies ξα||β||γ − ξα||γ||β = Rαρβγξρ in a metricspace (since the metric is used to lower indices, i.e. Tα

γ = gγβTαβ etc.). Also onehas generally

(B.3) Tαδ||β||γ − Tαδ

||γ||β = RατβγT τδ + Rδ

τβγTατ

379

380 APPENDIX B

The notation Rαηβγξη(α,β,γ) = 0 = Rαηβγ(α,β,γ)ξη involves an antisymmetriza-

tion in α, β, γ and since ξη is arbitrary this means Rαηβγ(α,β,γ) = 0. Writtenout (with some relabeling and combination) this means that there are symmetries

(B.4) Rαηβγ = −Rαηγβ , Rαηβγ = −Rηαβγ , Rαηβγ = Rβγαη

and R1023 + R2031 + R3012 = 0. The Bianchi identities are Rαηβγ||δ(β,γ,δ) = 0.

Next via parallel transport one has dξα = −

αβ γ

ξβdxγ and displacements

along paths dx, dx and dx, dx respectively leads to a vector transport difference∆ξα = Rα

βηγξβdxηdxγ . Now the only meaningful contraction of Rαβγδ is given byRηγ = Rα

ηαγ = gαβRβηαγ = gαβRαγβη = Rγη with 10 independent components.The equation Rβδ = 0 satisfies conditions 1-4 above and has the Lorentz metricfor one solution. Note here(B.5)

Rβδ =

αβ α

|δ−

αβ δ

|α

+

ατ δ

τ

β α

−

ατ α

τ

β δ

= 0

can be written out in terms of the metric tensor gαγ and this is the free spaceEinstein field equation. Some calculation shows that this can be written also interms of the zero divergence Ricci tensor Gβδ = Rβδ − (1/2)gβδR = 0 whereR = Rη

η is the Riemann scalar). Finally for a one parameter family Γ(v) ofgeodesics given by xµ = xµ(u, v) one has geodesic equations

(B.6)∂2xµ

∂u2= −

α

β γ

∂xβ

∂u

∂xγ

∂u

Now one can write out (B.5) in the form

(B.7) Rµν =12gρσ

[−gµσ|ν|ρ − gνρ|µ|σ + gµν|ρ|σ + gρσ|µ|ν

]+ Kµν

where Kµν contains only metric potentials and their first derivatives. One thinksof solving an initial value problem with data on a 3-dimensional hypersurfaceS described locally via x0 = 0 (so g00 > 0). On S one gives gαβ and all firstderivatives (only gµν and gµν|0 need to be prescribed). Then one can calculatethat

(B.8) Rij = (1/2)g00gij|0|0 + Mij = 0; Ri0 = −(1/2)g0jgij|0|0 + Mi0 = 0;

R00 = (1/2)gijgij|0|0 + M00 = 0where the Mµν can be computed from data on S. A change of coordinates willmake all gλ0|0|0 = 0 on S (these are not contained in (B.8)) and thus (B.8) consistsof 10 equations for six unknowns gij|0|0 on S which is overdetermined and leads tocompatability conditions for the data Mµν on S. This can be reduced to the form

(B.9) Rij = 0; G 0λ = 0

The first set of 6 equations determines the six unknowns gij|0|0 from initial data.The additional 4 equations in terms of the Ricci tensor G 0

λ represent necessaryconditions on the initial data in order to insure a solution. There is much morematerial in [12] about the Cauchy problem which we omit here. For the Einstein

RELATIVITY AND ELECTROMAGNETISM 381

equations in nonempty space one needs an energy momentum tensor for which atypical form is

(B.10) Tµν = ρ0uµuν + (p/c2)(uµuν − gµν)

where ρ0 is a density, p a pressure term, and uµ a 4-velocity field. One assumesTµν has zero divergence or Tµν

||ν = 0 (which is a covariant formulation of fluid flowunder the effect of its own internal pressure force).

REMARK B.1 One can also include EM fields in Tµν by dealing with aLorentz force f i = σ(E + (v/c) × H)i ∼ −σ0F

iνuν where F iν is the EM fieldtensor. However we will work in electromagnetism later in more elegant fashionvia the Dirac-Weyl theory and omit this here (cf. [12] for more details).

In any event the Einstein field equations for nonempty space involve a zerodivergence Tµν so one uses the Ricci tensor Gµν = Rµν − (1/2)gµνR and notesthat the most general second order tensor Bαγ of zero divergence can be writtenas Bαγ = Gαγ + Λgαγ (a result of E. Cartan). Hence one takes the Einstein fieldequations to be Gαγ + Λgαγ = cTαγ .

The exposition in Section 5.2 suggests the desirability of having a differentialform discription of EM fields and we supply this via [723]. Thus one thinks oftensors T = T σ

µν∂σ⊗dxµ⊗dxν with contractions of the form T (dxσ, ∂σ) ∼ Tνdxν .For η = ηµνdxµ ⊗ dxν one has η−1 = ηµν∂µ ⊗ ∂ν and ηη−1 = 1 ∼ diag(δµ

µ). Notealso e.g.

(B.11) ηµνdxµ ⊗ dxν(u,w) = ηµνdxµ(u)dxν(w) =

= ηµνdxµ(uα∂α)dxν(wτ∂τ ) = ηµνuµwν

(B.12) η(u) = ηµνdxµ ⊗ dxν(u) = ηµνdxµ(u)dxν =

= ηµνdxµ(uα∂α)dxν = ηµνuµdxν = uνdxν

for a metric η. Recall α ∧ β = α⊗ β − β ⊗ α and

(B.13) α ∧ β = αµdxµ ∧ βνdxν = (1/2)(αµβν − ανβµ)dxµ ∧ dxν

The EM field tensor is F = (1/2)Fµνdxµ ∧ dxν where

(B.14) Fµν =

⎛⎜⎜⎝0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

⎞⎟⎟⎠ ;

F = Exdx0 ∧ dx1 + Eydx0 ∧ dx2 + Ezdx0 ∧ dx3−−Bzdx1 ∧ dx2 + Bydx1 ∧ dx3 −Bxdx2 ∧ dx3

The equations of motion of an electric charge is then dp/dτ = (e/m)F(p) wherep = pµ∂µ. There is only one 4-form, namely ε = dx0 ∧ dx1 ∧ dx2 ∧ dx3 =(1/4!)εµνστdxµ∧dxν ∧dxσ ∧dxτ where εµνστ is totally antisymmetric. Recall alsofor α = αµν···dxµ ∧ dxν · · · one has dα = dαµν··· ∧ dxµ ∧ dxν · · · = ∂ααµν···dxσ ∧dxµ ∧ dxν · · · and ddα = 0. Define also the Hodge star operator on F and j via

382 APPENDIX B

∗F = (1/4)εµνστF στdxµ ∧ dxν and ∗j = (1/3!)εµνστ jτdxµ ∧ dxν ∧ dxσ; these arecalled dual tensors. Now the Maxwell equations are

(B.15) ∂µFµν =4π

cjν ; ∂αFµν + ∂µF να + ∂νFαµ = 0

and this can now be written in the form

(B.16) dF = 0; d∗F =4π

c∗ j

and 0 = d∗j = 0 is automatic. In terms of A = Aµdxµ where F = dA the relationdF = 0 is an identity ddA = 0.

A few remarks about the tensor nature of jµ and Fµν are in order and we writen = n(x) and v = v(x) for number density and velocity with charge density ρ(x) =qn(x) and current density j = qn(x)v(x). The conservation of particle numberleads to∇·j+ρt = 0 and one writes jν = (cρ, jx, jy, jz) = (cρn, qnvx, qnvy, qnvz) orequivalently jν = n0qu

ν ≡ jν = ρ0uν where n0 = n

√1− (v2/c2) and ρ0 = qn0 (ρ0

here is charge density). Since jν consists of uν multiplied by a scalar it must havethe transformation law of a 4-vector j

′β = aβνjν under Lorentz transformations.

Then the conservation law can be written as ∂νjν = 0 with obvious Lorentzinvariance. After some argument one shows also that Fµν = a ν

β a µα F

′αβ underLorentz transformations so Fµν is indeed a tensor. The equation of motion fora charged particle can be written now as (dp/dt) = qE + (q/c)v ×B where p =mv/

√1− (v2/c2) is the relativistic momentum. This is equivalent to dpµ/dt =

(q/m)pνFµν with obvious Lorentz invariance. The energy momentum tensor ofthe EM field is

(B.17) Tµν = −(1/4π)[FµαF να − (1/4)ηµνFαβFαβ ]

(cf. [723] for details) and in particular T 00 = (1/8π)(E2 +B2) while the Poyntingvector is T 0k = (1/4π)(E×B)k.

One can equally well work in a curved space where e.g. covariant derivativesare defined via

(B.18) ∇nT = limdλ→0[(T (λ + dλ)− T (λ)− δT ]/dλ

where δT is the change in T produced by parallel transport. One has then theusual rules ∇u(T ⊗ R) = ∇uT ⊗ R + T ⊗ ∇uR and for v = vν∂ν one finds∇µv = ∂µvν∂ν + vν∇µ∂ν . Now if v was constructed by parallel transport itscovariant derivative is zero so, acting with the dual vector dxα gives

(B.19)∂xν

∂xµdxα(∂ν) + vνdxα(∇µ∂ν) = 0 ≡ ∂µvα + vνdxα(∇µ∂ν) = 0

Comparing this with the standard ∂µvα + Γαµνvν = 0 gives dxα(∇µ∂ν) = Γα

µν .One can show also for vectors u, v, w (boldface omitted) and a 1-form α(B.20)(∇u∇v −∇v∇u − uv + vu)α(w) = R(α, u, v, w); R = Fσ

βµν∂σ ⊗ dxβ ⊗ dxµ ⊗ dxν

so R represents the Riemann tensor.


For the nonrelativistic theory first we go to [650] we define a transverse andlongitudinal component of a field F via

(B.21) F ||(r) = − 14π

∫d3r′

∇′ · F (r′)|r − r′| ; F⊥(r) =

14π∇×∇×

∫d3r′

F (r′)|r − r′|

For a point particle of mass m and charge e in a field with potentials A and φone has nonrelativistic equations mx = eE + (e/c)v × B (boldface is suppressedhere) where one recalls B = ∇ × A, v = x, and E = −∇φ − (1/c)At with H =(1/2m)(p− (e/c)A)2 + eφ leading to

(B.22) x =1

2m

(p− e

cA)

; p =e

c[v ×B + (v · ∇)A]− e∇φ

Recall here also

(B.23) B = ∇×A, ∇ · E = 0, ∇ ·B = 0, ∇× E = −(1/c)Bt,

∇×B = (1/c)Et, E = −(1/c)At −∇φ

(the Coulomb gauge∇·A = 0 is used here). One has now E = E⊥+E|| ∼ ET +EL

with ∇ · E⊥ = 0 and ∇ × E|| = 0 and in Coulomb gauge E⊥ = −(1/c)At andE|| = −∇φ. Further

(B.24) H ∼ 12m

(p− e

cA)2

+ eφ +18π

∫d3r((E⊥)2 + B2)

(covering time evolution of both particle and fields).

For the relativistic theory one goes to the Dirac equation i(∂t+α ·∇)ψ = βmψwhich, to satisfy E2 = p2 + m2 with E ∼ i∂t and p ∼ −i∇, implies −∂2

t ψ =(−iα · ∇ + βm)2ψ and ψ will satisfy the Klein-Gordon (KG) equation if β2 =1, αiβ + βαi ≡ αi, β = 0, and αi, αj = 2δij (note c = = 1 here withα · ∇ ∼

∑αµ∂µ and cf. [650, 632] for notations and background). This leads to

matrices

(B.25) σ1 =(

0 11 0

); σ2 =

(0 −ii 0

); σ3 =

(1 00 −1

);

αi =(

0 σi

σi 0

); β =

(1 00 −1

)where αi and β are 4 × 4 matrices. Then for convenience take γ0 = β and γi =βαi which satisfy γµ, γν = 2gµν (Lorentz metric) with (γi)† = −γi, (γi)2 =−1, (γ0)† = γ0, and (γ0)2 = 1. The Dirac equation for a free particle can now bewritten

(B.26)(

iγµ ∂

∂xµ−m

)ψ = 0 ≡ (i∂/−m)ψ = 0

where A/ = gµνγµAν = γµAµ and ∂/ = γµ∂µ. Taking Hermitian conjugates in,noting that α and β are Hermitian, one gets ψ(i

←−∂/ + m) = 0 where ψ = ψ†β. To

define a conserved current one has an equation ψγµ∂µψ +γµψµψ = ∂µ(ψγµψ) = 0

384 APPENDIX B

leading to the conserved current jµ = ψγµψ = (ψ†ψ, ψ†αψ) (this means ρ = ψ†ψand j = ψ†αψ with ∂tρ+∇· j = 0). The Dirac equation has the Hamiltonian form

(B.27) i∂tψ = −iα · ∇ψ + βmψ = (α · p + βm)ψ ≡ Hψ

(α · p ∼∑

αµpµ). To obtain a Dirac equation for an electron coupled to a pre-scribed external EM field with vector and scalar potentials A and φ one substitutespµ → pµ − eAµ, i.e. p→ p− eA and p0 = i∂t → i∂t − eΦ, to obtain

(B.28) i∂tψ = [α · (p− eA) + eΦ + βm]ψ

This identifies the Hamiltonian as H = α · (p−eA)+eΦ+βm = α ·p+βm+Hint

where Hint = −eα · A + eΦ, suggesting α as the operator corresponding to thevelocity v/c; this is strengthened by the Heisenberg equations of motion

(B.29) r =(

1i

)[r,H] = α; π =

(1i

)[π,H] = e(E + α×B)

Another bit of notation now from [647] is useful. Thus (again with c = = 1)one can define e.g. σz = −iαxαy, σx = −iαyαz, σy = −iαzαx, ρ3 = β, ρ1 =σzαz = −iαxαyαz, and ρ2 = iρ1ρ3 = βαxαyαz so that β = ρ3 and αk = ρ1σ

k.Recall also that the angular momentum of a particle is = r× p (∼ (−i)r×∇)with components k satisfying [x, y] = iz, [y, z] = ix, and [z, x] = iy. Anyvector operator L satisfying such relations is called an angular momentum. Nextone defines σµν = (1/2)i[γµ, γν ] = iγµγν (µ = ν) and Sαβ = (1/2)σαβ . Then the6 components Sαβ satisfy

(B.30) S10 = (i/2)αx; S20 = (i/2)αy; S30 = (i/1)αz;

S23 = (1/2)σx; , S31 = (1/2)σy; S12 = (1/2)σz

The Sαβ arise in representing infinitesimal rotations for the orthochronous Lorentzgroup via matrices I + iεSαβ . Further one can represent total angular momentumJ in the form J = L + S where L = r × p and S = (1/2)σ (L is orbital angularmomentum and S represents spin). We recall that the gamma matrices are givenvia γ = βα. Now from

(B.31) [(i∂t − eφ)− α · (−i∇− eA)− βm]ψ = 0

one gets

(B.32) [iγµDµ −m)ψ = [γµ(i∂µ − eAµ)−m]ψ = 0

where Dµ = ∂µ + ieAµ ≡ (∂0 + ieφ,∇− ieA). Working on the left with (−iγλDλ−m) gives then [γλγµDλDµ + m2]ψ = 0 where γλγµ = gλµ + (1/2)[γλ, γµ]. Byrenaming the dummy indices one obtains

(B.33) [γλ, γµ]DλDµ = −[γλ, γµ]DµDλ = (1/2)[γλ, γµ][Dλ, Dµ]

leading to

(B.34) [Dλ, Dµ] = ie[∂λ, Aµ] + ie[Aλ, ∂µ] = ie(∂λAµ − ∂µAλ) = ieFλµ

This yields then

(B.35) γλγµDλDµ = DµDλ + eSλµFλµ


where Sλµ represents the spin of the particle. Therefore [DµDµ + eSλµFλµ +m2]ψ = 0. Comparing with the standard form of the KG equation we see thatthis differs by the term eSλµFλµ which is the spin coupling of the particle to theEM field and has no classical analogue.

APPENDIX C

REMARKS ON QUANTUM GRAVITY

We refer here to [55, 56, 57, 58, 60, 61, 62, 69, 70, 184, 303, 394, 495,551, 630, 665, 819, 820, 896, 897, 898, 929, 930, 931, 932] and will mainlyfollow [69, 551] for basic material. First (cf. [69] recall that upon assuming thespacetime manifold M to be diffeomorphic to R × S where S is a 3-dimensionalmanifold one can choose spacelike slices Σ ⊂M with a space Met(Σ) of Riemann-ian metrics. Writing φ : M → R×S one defines a time coordinate via τ = φ∗t andΣ ⊂ M is determined via τ = constant. The extrinsic curvature K of Σ providesCauchy data (3g,K) for the metric and regarding Einstein’s 10 equations, 4 areconstraint equations for the Cauchy data and 6 are evolution equations saying howthe 3-metric changes in time. This is the Arnowitt-Deser-Misner (ADM) formula-tion. Now in more detail, one takes g(v, v) > 0 for v ∈ TΣ and g(n, n) = −1 (wheren is the unit normal vector to Σ). One can write v = −g(v, n)n+(v + g(v, n)n) interms of orthogonal vectors and, for ∇ corresponding to the covariant derivativefor the Levi-Civita connection, one can write

(C.1) ∇uv = −g(∇uv, n) + (∇uv + g(∇uv, n)n); K(u, v)n = −g(∇uv, n)n

where K defines the extrinsic curvature. Since ∇ is torsion free one has K(u, v) =K(v, u) and

(C.2) Kijuivj = K(∂i, ∂j)uivj ; K(u, v) = g(∇un, v)

Write now ∂τ = Nn + N whee N is the shift field and N the lapse function. Thenone has

(C.3) N = −g(∂τ , n); N = ∂τ + g(∂τ , n)n

Write the Christoffel symbols of the connection ∇ on Σ as 3Γijk and the Riemann

tensor of 3g as 3Rmijk. Then some calculation (cf. [69]) gives the Gauss-Codazzi

equations

(C.4) R(∂i, ∂j)∂k = (3∇iKjk − 3∇jKik)n + (3Rmijk + KjkKm

i −KikKmj )∂m

Now the Einstein tensor has the form Gµν = Rµν− (1/2)gµνR where Rµν = Rαµαν

with R = Rαβαβ and for vectors ∂j tangent to Σ at a point in question this leads

to (cf. [69]

(C.5) Gµνnµnν = −12(3R + (TrK)2 − Tr(K2)); Gνin

µ = 3∇jKji − 3∇iK

jj

387

388 APPENDIX C

The remaining 6 Einstein equations Gij = 0 are dynamical in nature and describethe time evolution of 3g. Now looking only at the vacuum Einstein equations onewrites qij ∼ 3gij with q for det(qij). It can be shown then (cf. [69]) that

(C.6) Kij =12N−1(qij − 3∇iNj − 3∇jNi)

The Lagrangian density for the Einstein-Hilbert action is R√−detgd4x and one

writes here L = R√−g which in terms of the 3-metric and lapse function becomes

L = q1/2NR. Discarding terms that give total divergences (and would integrateto zero for compact Σ at least) one has then

(C.7) L = q1/2N(3R + Tr(K2)− (trK)2)

Now the conjugate momenta are determined via

(C.8) pij =∂L

∂qij= q1/2(Kij − Tr(K)qij)

The Hamiltonian structure involves now

(C.9) H(pij , qij) = pij qij − L; H =

∫Hd3x; H = q1/2(NC + N iCi)

where

(C.10) C = −3R + q−1

(Tr(p2)− 1

2Tr(p)2

); Ci = −2 3∇j(q−1/2pij)

Note one must specify the lapse and shift to know the meaning of time evolution.However one can compute that

(C.11) C = −2Gµνnµnν ; Ci = −2Gµinµ

Now the equations C = Ci = 0 are precisely the constraint Einstein equations andhence H = 0 is a constraint on the phase space in T ∗Met(Σ) yet the dynamics isnot trivial. To formulate Hamilton’s equations one can define Poisson brackets onphase space via

(C.12) f, g =∫

Σ

∂f

∂pij(x)∂g

∂qij(x)− ∂f

∂qij(x)∂g

∂pij(x)

q1/2d3x

This is often written in a functional derivative notation but it simply amounts todefining say ∂f/∂qij(x) for example via

(C.13)∫

Σ

hij(x)∂f

∂qij(x)q1/2d3x =

d

dsf(q + sh)|s=0

for every symmetric (0, 2) tensor field h (for h ∼ δg one writes the right side of(C.13) as δf . In this spirit one arrives at

(C.14) pij(x), qk(y) = (δikδj

+ δiδ

jk)δ3(x− y);

pij(x), pk(y) = 0; qij(x), qk(y) = 0

REMARKS ON QUANTUM GRAVITY 389

The equations Gij = 0 in this disguise are now simply qij = H, qij and pij =H, pij which take the form

(C.15) qij = 2q−1/2N

(pij −

12pk

kqij

)+ 2 3∇[iNj];

pij = −Nq1/2

(3Rij − 1

23Rqij

)+

12Nq−1/2qij

(pabp

ab −−12(pa

a)2)−

−2Nq−1

(piapj

a −12pa

aqij

)+ q1/2(∇i∇jN − qij 3∇a 3∇aN)+

+q1/2∇a(q−1/2Napij)− 2pa[i3∇aN j]

This is a horror story but it does show that the time evolution given by Hamilton’sequations is nontrivial. One notes in passing that the lapse and shift measurehow much the time evolution push the slice Σ in the normal or tangent directionrespectively. For example for shift (resp. lapse) zero one has

(C.16) C(N) =∫

Σ

NCq1/2d3x (resp. C( N) =∫

Σ

N iCiq1/2d3x)

Here C( N) or Ci (resp. C(N) or C) is called the diffeomorphism (resp. Hamil-tonian) constraint and one can calculate

(C.17) C( N), C( N ′) = C([ N, N ′]); C( N), C(N ′) = C( NN ′);

C(N), C(N ′) = C((N∂iN ′ −N ′∂iN)∂i)

where NN ′ is the derivative of N ′ in the direction N and (N∂iN ′ −N ′∂iN)∂i isthe result of converting the 1-form NdN ′ − N ′dN into a vector field by raisingindices. These formulas are known as the Dirac algebra and one notes that theconstraints are closed under Poisson brackets.

Now for quantization one proceeds formally for various reasons (cf. [69]) andtakes the operator corresponding to the 3-metric (and momentum) to be

(C.18) (qij(x)ψ)(q) = gij(x)ψ(q); (pij(x)ψ)(q) = −i∂

∂qij(x)ψ(q)

where g ∈ Met(Σ). These operators satisfy then

(C.19) [pij(x), qk(y)] = −i(δikδj

+ δiδ

jk)δ3(x, y);

[pij(x), pk(y)] = 0; [qij(x), qk(y)] = 0

In order to obtain quantum versions C and Ci one encounters operator orderingproblems; ideally one would like

(C.20) [C( N), C( N ′)] = −iC( N, N ′]); [C( N, C(N ′)] = −iC( NN ′);

[C(N), C(N ′)] = −iC((N∂iN ′ −N ′∂iN)∂i)but this seems virtually impossible to achieve. Suppose nevertheless that oneobtained somehow satisfactory operators C and Ci; then one could write

(C.21) H =∫

Σ

(NC + N iCi)q1/2d3x

390 APPENDIX C

It could then be said that a vector ψ ∈ L2(Met(Σ)) (whatever that may mean)is a physical state if it satisfies C(N)ψ = C( N)ψ = 0 or alternatively Hψ = 0for all choices of lapse and shift and this is the WDW equation. There are manyproblems here and even if one could find solutions there arises the problem of time,namely the Hamiltonian vanishes on the space of physical states so any operatorA on Hphys must satisfy (d/dt)At = i[H, At] = 0 and the dynamics disappears.Recent developments using the Ashtekar variables have made some progress in thisarea and will be discussed briefly below.

We describe briefly now the Ashtekar variables following [69] which are basedon a modification of the Palatini formalism. First for background the Palatiniaction is S(g) =

∫M

R · vol rewritten so that it is not a function of the metricbut rather a function of a connection and a frame field. Thus a trivializationof TM is a vector bundle isomorphism e : M × Rn → TM sending each fiberp ×Rn of the trivial bundle M ×Rn to the corresponding tangent space TpM .A trivialization of TM is also called a frame field sending the standard basis ofRn to a basis of tangent vectors at p or a frame. If M is 3 (resp. 4) dimensionala frame field is called a triad (resp. tetrad). One goes back and forth now usingthe frame field e and its inverse e−1 : TM → M ×Rn. Given a basis of sectionsof M × Rn of the form ξi = (0, · · · , 0, 1, 0, 0, · · · ) one writes any section as s =sIξI where I denotes an internal index (whereas ∂µ refers to coordinate vectorfields on a chart). Thus e(ξI) = eα

I ∂α and e(ξI) ∼ eI . Now given sections s, s′

one can define η(s, s′) = ηIJsIs′J where ηIJ is copied after a Minkowski metric

(−1, 1, · · · , 1) (internal metric). One can raise and lower indices via ηIJ and setg(v, v′) = gαβvαv

′β . The frame field is said to be orthonormal if g(eI , eJ ) = ηIJ ;in this case one has g(e(s), e(s′)) = η(s, s′) since

(C.22) g(e(s), e(x′)) = g(e(SIξI), e(sJξJ)) = sIsJg(eI , eJ ) = ηIJsIsJ =

= η(sIξI , sJξJ ) = η(s, s′)

Note that one can write ηIJ = g(eI , eJ ) = gαβeαI eβ

J and hence δIJ = eI

αeαJ . Further

e−1v = eIαvαξI since if v = e(s) one has e−1v = eI

αvαξI = eIαeα

JsJξI = δIJsJξI =

sIξI = s This leads to a formula for the metric g in terms of the coframe field eIα,

namely

(C.23) gαβ = g(∂α, ∂β) = η(e−1∂α, e−1∂β) = η(eIαξI , e

JβξJ) = ηIJeI

αeJβ

The other ingredient in the Palatini formalism is a connection on the trivial bun-dle M ×Rn. One says a connection D here is a Lorentz connection if vη(x, s′) =η(Dvs, s′)+η(s, Dvs′) (standard S(O(n, 1) connection). Note torsion free is mean-ingless and there is no Levi-Civita connection on M×Rn; however the standard flatconnection D0s = v(sI)ξI is nice and any connection can be written as D = D0+Afor some potential A, which is an End(Rn)-valued 1-form on M. Thus

(C.24) Dvs = (v(sJ) + AJµIv

µsI)ξJ ; F IJαβ = ∂αAIJ

β − ∂βAIJα + [Aα, Aβ ]IJ

where F IJαβ is the curvature of D. If A defines a Lorentz connection then F IJ

αβ =−F IJ

βα = −F JIαβ . Next given a frame field e and a Lorentz connection D one can


transfer the Lorentz connection from M ×RN to TM to obtain a connection ∇given by

(C.25) ∇α∂β = Γγαβ∂γ ; Γγ

αβ = AJαIe

Iβeγ

J

Here ∇ is called the imitation Levi-Civita connection and Γγαβ are the imitation

Christoffel symbols, leading to an imitation Riemann tensor

(C.26) R δαβ = F IJ

αβeγI eδ

J ; Rαβ = Rγαγβ ; R = Rα

α

Now the Palatini action is basically the Einstein-Hilbert action in disguise, beinga function of the frame field given by gαβ = ηIJeI

αeJβ in the form

(C.27) S(A, e) =∫

M

eαI eβ

JF IJαβ · vol; δS = 2

∫M

(Rαβ − (1/2)RgαβηIJeβJ (δeα

I ) · vol

Thus δS = 0 for an arbitrary variation of the frame field when Rαβ−(1/2)Rgαβ = 0which is of course Einstein’s equation when ∇ = ∇. We refer to [69] for furthercomputations, formulas, and discussion.

Now for the Ashtekar variables themselves define the complexified tangentbundle CTM to have fibers C × TpM and an imitation complexified tangentbundle M ×C4. A complex frame field is an isomorphism e : M ×C4 → CTM .A connection A on M ×C4 is an End(C4) valued 1-form on M with componentsAJ

αI or AIJα ; it is Lorentz if AIJ

α = −AJIα . Recall the Hodge ∗ operator maps

2-forms to 2-forms in 4 dimensions and define it here via

(C.28) ∗T IJ = (1/2)εIJKLTKL; εi1,··· ,in =

sgn(i1, · · · , in) ij distinct

0 otherwise

for any T with two antisymmetric raised internal indices. In particular

(C.29) (∗A)IJα = (1/2)εIJ

KLAKLα

Now write any Lorentz connection A as a sum of self-dual and anti-self-dual parts

(C.30) A = +A + −A; ∗±A = ±i±A; ±A = (1/2)(A∓ i ∗A)

In the self dual formulation of GR one of the two basic fields is a self-dual Lorentzconnection, i.e. a Lorentz connection on M × C4 with ∗+A = i+A. The otherbasic field is a complex frame field e : M × C4 → CTM and the action is builtusing the curvature +F of +A via

(C.31) +F IJαβ = ∂α

+AIJβ − ∂β

+AIJα + [+Aα, +Aβ ]IJ

As in the Palatini formalism one writes

(C.32) gαβ = ηIJeIαeJ

β ; e−1∂α = eIαξI

(note the metric is now complex). The self dual action is

(C.33) SSD(+A, e) =∫

M

eαI eβ

J+F IJ

αβ · vol; vol =√−gd4x

392 APPENDIX C

Now define the internal Hodge dual of the curvature of a connection F on M ×C4

via

(C.34) (∗F )IJαβ = (1/2)εIJ

KLFKLαβ

and call the curvature self dual if ∗F = iF . It turns out that the curvature ofa self dual Lorentz connection is self dual (computation needed) and this has Liealgebraic meaning (cf. [69] for details). Next compute δSSD = 0 and following[69] one obtains two equations. First by varying the self dual connection thereis an equation saying that +A is the self dual part of a Lorentz connection A forwhich the self dual part of the Riemann tensor of g is related to +F , namely

(C.35) +Rα δβγ = (1/2)(Rα δ

βγ − (i/2)εαδµνRµ ν

βγ ); +Rα δβγ = +F IJ

βγ eαI eδ

J

Second by varying the frame field there arises a self dual analogue of Einstein’sequation

(C.36) +Rαβ − (1/2)gαβ+R = 0; +Rαβ = +Rγ

αγβ ; +R = +Rαα

Using symmetries of the Riemann tensor this is equivalent to the vacuum Einsteinequation. Note however that we have complex metrics here; some reality conditionsare needed and this is not exactly a trivial matter. Thus let Σ be a spacelikeslice and work in coordinates such that ∂0 is normal to Σ and ∂i is tangent forspacelike indices. Given a self dual Lorentz connection +AIJ

α on M ×C4 one canrestrict it to a connection AIJ

i on Γ × C4 with AIJi = −AJI

i and ∗A = iA (the+ sign is gratuitously omitted here). Since sl(2,C) has a basis in terms of Paulimatrices one can also write this as −(i/2)Aa

i σa (a = 1, 2, 3). The field playing therole analogous to position is the self dual Lorentz conection Aa

i with conjugatemomentum Ei

a = q1/2eia and one has Poisson brackets

(C.37) Eia(x), Ab

j(y) = −iδbaδi

jδ3(x, y); Ei

a(x), Ejb (y) = 0 = Aa

i (x), Abj(y)

The Hamiltonian and diffeomorphism constraints are given via

(C.38) C = εabcEiaEj

bFijc; Cj = EkaF a

jk; Ga = DiEia

(the latter being a Gauss law constraint). The tilde appears because of densitation,i.e. C is q1/2 times the earlier C. To quantize now one writes

(C.39) (Aai (x)ψ)(A) = Aa

i (x)ψ(A); (Eia(x)ψ)(A) =

∂

∂Aai (x)

ψ(A)

(C.40) [Eia(x), Ab

j(y)] = δbaδi

jδ3(x, y); [Ei

a(x), Ejb (y)] = 0 = [Aa

i (x), Abj(y)]

A convenient choice of operator orderings is now(C.41)

C = εabcEiaEj

b Fijc; Cj = Eka F a

jk; Ga = DiEia; (F a

jk(x)ψ)(A) = F ajk(x)ψ(A)

It seems that these operators satisfy commutation relations analogous to the Pois-son brackets of the classical constraints. The physical state space Hphys thenconsists of functions ψ(A) satisfying the constraints in quantum form, i.e.

(C.42) Hphys = ψ : Cψ = Cjψ = Gaψ = 0


Here Cjψ = 0 means ψ(A) = ψ(A′) whenever A′ is obtained from A by applying adiffeomorphism connected to the identity by a flow. Similarly Gaψ = 0 says thatψ(A) = ψ(A′) whenever A′ is obtained from A by a small gauge transformation.The Hamiltonian constraint Cψ = 0 contains the dynamics of the theory and find-ing solutions is difficult.

There is an interesting relation between Chern-Simons (CS) theory and quan-tum gravity however that provides some solutions. First if the cosmological con-stant is nonzero the Hamiltonian constraint becomes

(C.43) C = εabcEiaEj

b Fijc −Λ6

εijkεabcEiaEj

b Ekc

The CS state ψCS is defined as ψCS(A) = exp[−(6/)SCS(A)] where SCS(A) =∫Σ

Tr(A∧dA+(2/3)A∧A∧A). It is then shown in [69] that CjψCS = GaψCS =CψCS = 0 with some discussion. The book [69] was written in 1994 and therehas since been enormous activity in loop quantum gravity for which we refer to[53, 57, 60, 184, 303, 394, 551, 819, 820, 896, 929, 930, 931, 932]. (cf. also[714 ] for connections between thermodynamics and gravity).

APPENDIX D

DIRAC ON WEYL GEOMETRY

First we give some background on Weyl geometry and Brans-Dicke theoryfollowing [12]; for differential geometry we use the tensor notation of [12] andrefer to e.g. [149, 358, 458, 723, 731, 972, 998] for other notation (see also[990] for interesting variations). For general background see [156, 179, 235, 350,351, 361, 556, 601, 786, 801, 937, 938, 939, 1004, 1014] and note that forour purposes the most important background features appear already in Sections3.2, 3.2.2, and 4.1. One thinks of a differential manifold M = Ui, φi withφ : Ui → R4 and metric g ∼ gijdxidxj satisfying g(∂k, ∂) = gk =< ∂k, ∂ >=gk. This is for the bare essentials; one can also imagine tangent vectors Xi ∼∂i and dual cotangent vectors θi ∼ dxi, etc. Given a coordinate change xi =xi(xj) a vector ξi transforming via ξi =

∑∂ix

jξj is called contravariant (e.g.dxi =

∑∂j x

idxj). On the other hand ∂φ/∂xi =∑

(∂φ/∂xj)(∂xj/∂xi leads tothe idea of covariant vectors Aj ∼ ∂φ/∂xj transforming via Ai =

∑(∂xj/∂xi)Aj

(i.e. ∂/∂xi ∼ (∂xj/∂xj)∂/∂xj). Now define connection coefficients or Christoffelsymbols via (strictly one writes T γ

α = gαβT γβ and T γα = gαβT βγ which are

generally different; we use that notation here but it is not used in subsequentsections since it is unnecessary)

(D.1) Γrki = −

r

k i

= −1

2

∑(∂igk + ∂kgi − ∂gik)gr = Γr

ik

(note this differs by a minus sign from some other authors). Note also that (D.1)follows from equations

(D.2) ∂gik + grkΓri + girΓr

k = 0

and cyclic permutation; the basic definition of Γimj is found in the transplantation

law

(D.3) dξi = Γimjdxmξj

Next for tensors Tαβγ define derivatives

(D.4) Tαβγ|k = ∂kTα

βγ ; Tαβγ|| = ∂T

αβγ − Γα

sTsβγ + Γs

βTαsγ + Γs

γTαβs

In particular covariant derivatives for contravariant and covariant vectors respec-tively are defined via

(D.5) ξi||k = ∂kξi − Γi

kξ = ∇kξi; ηm|| = ∂ηm + Γr

mηr = ∇ηm

395

396 APPENDIX D

Now to describe Weyl geometry one notes first that for Riemannian geometry(D.3) holds along with

(D.6) 2 = ‖ξ‖2 = gαβξαξβ ; ξαηα = gαβξαηβ

However one does not demand conservation of lengths and scalar products underaffine transplantation (D.3). Thus assume

(D.7) d = (φβdxβ)

where the covariant vector φβ plays a role analogous to Γαβγ . Combining one

obtains

(D.8) d2 = 22(φβdxβ) = d(gαβξαξβ) =

= gαβ|γξαξβdxγ + gαβΓαργξρξβdxγ + gαβΓβ

ργξαξρdxγ

Rearranging etc. and using (D.6) again gives

(D.9) (gαβ|γ − 2gαβφγ) + gσβΓσαγ + gσαΓσ

βγ = 0

leading to

(D.10) Γαβγ = −

α

β γ

+ gσα[gσβφγ + gσγφβ − gβγφσ]

Thus we can prescribe the metric gαβ and the covariant vector field φγ and de-termine by (D.10) the field of connection coefficients Γα

βγ which admits the affinetransplantation law (D.3). If one takes φγ = 0 the Weyl geometry reduces to Rie-mannian geometry. This leads one to consider new metric tensors via the gaugetransformation gαβ = f(xλ)gαβ and it turns out that (1/2)∂log(f)/∂xλ plays the

role of φλ in (D.7). The ordinary connections

αβ γ

constructed from gαβ

are equal to the more general connections Γαβγ constructed according to (D.10)

from gαβ and φλ = (1/2)∂log(f)/∂xλ. The generalized differential geometry isconformal in that the ratio

(D.11)ξαηα

‖ξ‖‖η‖ =gαβξαηβ

[(gαβξαξβ)(gαβηαηβ)]1/2

does not change under the gauge transformation above. Again if one has a Weylgeometry characterized by gαβ and φα with connections determined by (D.10) onemay replace the geometric quantities by use of a scalar field f with

(D.12) gαβ = f(xλ)gαβ , φα = φα + (1/2)(log(f)|α; Γαβγ = Γα

βγ

without changing the intrinsic geometric properties of vector fields; the only changeis that of local lengths of a vector via 2 = f(xλ)2. Note that one can reduceφα to the zero vector field if and only if φα is a gradient field, namely Fαβ =φα|β − φβ|α = 0 (i.e. φα = (1/2)∂alog(f) ≡ ∂βφα = ∂αφβ). In this case one haslength preservation after transplantation around an arbitrary closed curve and thevanishing of Fαβ guarantees a choice of metric in which the Weyl geometry becomes

DIRAC ON WEYL GEOMETRY 397

Riemannian; thus Fαβ is an intrinsic geometric quantity for Weyl geometry - noteFαβ = −Fβα and

(D.13) Fαβ|γ = 0; Fµν|λ = Fµν|λ + Fλµ|ν + Fνλ|µ

Similarly the concept of covariant differentiation depends only on the idea of vectortransplantation. Indeed one can define

(D.14) ξα||β = ξα

|β − Γαβγξγ

In Riemann geometry the curvature tensor is

(D.15) ξα||β|γ − ξα

||γ|β = Rαηβγξη

Hence here we can write

(D.16) Rαβγδ = −Γα

βγ|δ + Γαβδ|γ + Γα

τδΓτβγ − Γα

τγΓτβδ

Using (D.11) one then can express this in terms of gαβ and φα but this is compli-cated. Equations for Rβδ = Rα

βαδ and R = gβδRβδ are however given in [12]. Onenotes that in Weyl geometry if a vector ξα is given, independent of the metric, thenξα = gαβξβ will depend on the metric and under a gauge transformation one hasξα = f(xλ)ξα. Hence the covariant form of a gauge invariant contravariant vectorbecomes gauge dependent and one says that a tensor is of weight n if, under agauge transformation Tα···

β··· = f(xλ)nTα···β··· . Note φα plays a singular role in (D.12)

and has no weight. Similarly (√−g = f2√−g (weight 2) and Fαβ = gαµgβνFµν

has weight −2 while ,Fαβ = Fαβ√−g has weight 0 and is gauge invariant; furtherFαβFαβ√−g is gauge invariant. Now for Weyl’s theory of electromagnetism onewants to interpret φα as an EM potential and one has automatically the Maxwellequations Fαβ|γ = 0 along with a gauge invariant complementary set F

αβ|β = sα

(source equations). These equations are gauge invariant as a natural consequenceof the geometric interpretation of the EM field. For the interaction between theEM and gravitational fields one sets up some field equations as indicated in [?]and the interaction between the metric quantities and the EM fields is exhibitedthere.

REMARK 5.3.1. As indicated earlier in [12] Rijk is defined with a minus

sign compared with e.g. [723, 998]. There is also a difference in definition of theRicci tensor which is taken to be Gβδ = Rβδ − (1/2)gβδR in [12] with R = Rδ

δ

so that Gµγ = gµβgγδGβδ = Rµγ − (1/2)gµγR with Gγ

η = Rηη − 2R ⇒ Gη

η = −R(recall n = 4). In [723] the Ricci tensor is simply Rβµ = Rα

βµα where Rαβµν

is the Riemann curvature tensor and R = Rηη again. This is similar to [998]

where the Ricci tensor is defined as ρj = Riji. To clarify all this we note that

Rηγ = Rαηαγ = gαβRβηαγ = −gαβRβηγα = −Rα

ηγα which confirms the minus signdifference.

For completeness it is worthwhile to reflect on the comments of a mastercraftsman and hence we refer her to [302] where first there are two papers on anew classical theory of the electron and in the third paper of [302] the originalDirac-Weyl action is developed (cf. also Sections 3.2.1 and 4.1) which we sketch

398 APPENDIX D

here in some detail. The main point is to think of EM fields as a property of space-time rather than something occuring in a gravity formed spacetime. This seemsto be in the spirit of considering a microstructure of the vacuum (or an ether) andwe find it attractive. The solution proposed by Weyl involved a length changeδ = κµδxµ under parallel transport xµ → xµ + δxµ. The κµ are field quantitiesoccuring along with the gµν in a fundamental role. Suppose gets changed to′ = λ(x) and + δ becomes

(D.17) ′ + δ′ = ( + δ)λ(x + δx) = ( + δ)λ(x) + λ,µδxµ

with neglect of second order terms (here λ,µ ≡ ∂λ/∂xµ). Then

(D.18) δ′ = λδ + λ,µδxµ = λ(κµ + φ,µ)δxµ; φ = log(λ)

Hence

(D.19) δ′ = ′κ′µδxµ; κ′

µ = κµ + φ,µ

If the vector is transported by parallel displacement around a small closed loopthe total change in length is

(D.20) δ = FµνδSµν ; Fµν = κµ,ν − κν,µ

and δSµν is the element of area enclosed by the small loop. this change is un-affected by (D.19). It will be seen that the field quantities κµ can be taken tobe EM potentials, subject to the transformations (D.19) which correspond to nochange in the geometry but a change only in the choice of artificial standards oflength. The derived quantities Fµν have a geometrical meaning independent ofthe length standard and correspond to the EM fields. Thus the Weyl geometryprovides exactly what is needed for describing both gravitational and EM fields ingeometric terms. There was at first some apparent conflict with atomic standardsand the theory was rejected, leaving only the idea of gauge transformation forlength standard changes.

Dirac’s approach however helped to resurrect the Weyl theory; since we feelthat this theory is not perhaps sufficiently appreciated a sketch is given here (cf.also [121]). Dirac first goes into a discussion of large numbers, e.g. e2/GMm(proton and electron masses), e2/mc2 (age of universe), etc. and the Einsteiniantheory requires that G be constant which seems in contradiction to G ∼ t−1 wheret represents the epoch time, assumed to be increasing. Dirac reconciles this byassuming the large numbers hypothesis (all dimensionless large numbers are con-nected) and stipulating that the Einstein equations refer to an interval dsE whichis different from the interval dsA measured by atomic clocks. Then the objectionsto Weyl’s theory vanish and it is assumed to refer to dsE . In this spirit then onedeals with transformations of the metric gauge under which any length such asds is multiplied by a factor λ(x) depending on its position x, i.e. ds′ = λds anda localized quantity Y may get transformed according to Y ′ = λnY , in whichcase Y is said to be of power n and is called a co-tensor. If n = 0 then Y iscalled an in-tensor and it is invariant under gauge transformations. The equa-tion ds2 = gµνdxµdxν shows that gµν is a co-tensor of power 2, since the dxµ arenot affected by a gauge transformation. Hence gµν is a co-tensor of power −2 and


one writes g for√−g with T:µ denoting the covariant derivative (∇µT would be

better). We see that the covariant derivative of a co-tensor is not generally a co-tensor. However there is a modifed covariant derivative T∗µ which is a co-tensor.Consider first a scalar S of power n; then S:µ = S,µ ≡ Sµ; under a change of gaugeit transforms to

(D.21) S′µ = (λnS),µ = λnSµ + nλn−1λµS = λn[Sµ + n(κ′

µ − κµ)S]

(via (D.19)). Thus

(D.22) (Sµ − nκµS)′ = λn(Sµ − nκµS)

so Sµ−nκµS is a covector of power n and is defined to be the co-covariant derivativeof S, i.e.

(D.23) §∗µ = Sµ − nκµS

To obtain the co-covariant derivative of co-vectors and co-tensors we need a mod-ified Christoffel symbol

(D.24) ∗Γαµν = Γα

µν − gαµκν − gα

ν κµ + gµνκα

(the notation Γαµν for the more correct form Γα

µν is used in [302]). This is knownto be invariant under gauge transformations. Let now Aµ be a co-vector of powern and form Aµ,ν − ∗Γα

µνAα which is evidently a tensor since it differs from thecovariant derivative Aµ:ν by a tensor and under gauge transformations one has(cf. (D.19) where φ,µ = κ′

µ − κµ)

(D.25) (Aµ,ν − ∗ΓαµνAα)′ = λnAµ,ν + nλn−1λνAµ − ∗Γα

µνλnAα =

= λn[Aµ,ν + n(κ′ν − κν)Aµ − ∗Γα

µνAα]Thus

(D.26) (Aµ,ν − nκνAµ − ∗ΓαµνAα)′ = λn[Aµ,ν − nκνAµ − ∗Γα

µνAα]

so take

(D.27) Aµ∗ν = Aµ,ν − nκνAµ − ∗Γαµν

as the co-covariant derivative of Aα. this can be written via (D.24) as

(D.28) Aµ∗ν = Aµ:ν − (n− 1)κνAµ + κµAν − gµνκαAα

Similarly for a vector Bµ of power n one has

(D.29) Bµ∗ν = Bµ

:ν − (n + 1)κνBµ + κµBν − gµν καBα

For a co-tensor with various suffixes up and down one can form the co-covariantderivative via the same rules; one notes that the co-covariant derivative always hasthe same power as the original. Next observe

(D.30) (TU)∗σ = T∗σU + TU∗σ

while

(D.31) gµν∗σ = 0; Gµν∗σ = 0

so one can raise and lower suffixes freely in a co-tensor before carrying out co-covariant differentiation. Thus one can raise the µ in (D.28) giving (D.29) withAµ replacing Bµ and n−2 in place of n. The potentials κµ do not form a co-vector

400 APPENDIX D

because of the wrong transformation laws (D.19) but the Fµν defined by (D.19)are unaffected by gauge transformations so they form an in-tensor. One obtainsthe co-covariant divergence of a co-vector Bµ by putting ν = µ in (D.29) to get

(D.32) Bµ∗µ = Bµ

:µ − (n + 4)κµBµ

(for n = −4 this is the ordinary covariant divergence).

We list some formulas for second co-covariant derivatives now with a sketchof derivation. Thus for a scalar of power n

(D.33) S∗µ∗ν = S∗µ:ν − (n− 1)κνS∗µ + κµS∗ν − gµνκσS∗σ

Putting S∗µ = Sµ − nκµS on gets

(D.34)S∗µ∗ν = Sµ:ν − nκµ:ν − nκµSν − nκν(Sµ − nκµS) + κνS∗µ + κµS∗ν − gµνκσSσ

Now Sµ:ν = Sν:µ so

(D.35) S∗µ∗ν − S∗ν∗µ = −n(κµ:ν − κν:µ)S = −nFµνS

This is tedious but instructive and we continue. Let Aµ be a co-vector of power nso

(D.36) Aµ∗ν∗σ = Aµ∗ν:σ − nκσAµ∗ν + (gρµκσ + gρ

σκµ − gµσκρ)Aρ∗ν+

+(gρνκσ + gρ

σκν − gσνκρ)Aµ∗ρ

A lengthy calculation then yields

(D.37) Aµ∗ν∗σ −Aµ∗σ∗ν = ∗BµνσρA

ρ − (n− 1)FνσAµ

where

(D.38)∗Bµνσρ = Bµνσρ + gρν(κµ:σ + κµκσ) + gµσ(kρ:ν + κρκν)− gρσ(κµ:ν + κµκν)−

−gµν(κρ:σ + (κρκσ) + (gρσgµν − gρνgµσ)κακα

One can consider ∗B as a generalized Riemann-Christoffel tensor but it does nothave the usual symmetry properties for such a tensor; however one can write

(D.39) ∗Bµνσρ = ∗Bµνσρ + (1/2)(gρνFµσ + gµσFρν − gρσFµν − gµνFρσ)

and then ∗Bµνσρ has all the usual symmetries, namely

(D.40)∗Bµνσρ = −∗Bµσνρ = −∗Bρνσµ = ∗Bνµρσ; ∗Bµνσρ + ∗Bµσρν + ∗Bµρνσ = 0

Thus is is appropriate to call ∗Bµνσρ the Riemann-Christoffel (RC) tensor for Weylspace; it is a co-tensor of power 2. The contracted RC tensor is

(D.41) ∗Rµν = ∗Bσµνσ = Rµν − κµ:ν − κν:µ − gµνκσ

:σ − 2κµκν + 2gµνκσκσ

and is an in-tensor. A further contraction gives the total curvature

(D.42) ∗R = ∗Rσσ = R− 6κσ

:σ + 6κσσ

which is a co-scalar of power -2.

One gets field equations from an action principle with an in-invariant action,


hence one of the form I =∫

Ωgd4x where Ω must be a co-scalar of power -4 tocompensate g having power 4. Ths usual contribution to Ω from the EM field is(1/4)FµνFµν (of power -4 since it can be written as FµνFρσgµρgνσ with F factorsof power zero and g factors of power -2). One also needs a gravitational term andthe standard −R could be ∗R but this has power -2 and will not do. Weyl proposed(∗R)2 which has the correct power but seems too complicated to be satisfactory.Here one takes ∗R = 0 as a constraint and puts the constraint into the Lagrangianvia γ∗R with γ a co-scalar field of power -2 in the form of a Lagrange multiplier.This leads to a scalar-tensor theory of gravitation and one can insert other termsinvolving γ. For convenience one takes γ = −β2 with β as the basic field variable(co-scalar of power -1) and adds terms kβ∗σβ∗σ (co-scalar of power -4); terms cβ4

can also be added to get

(D.43) I =∫

[(1/4)FµνFµν − β2∗R + kβ∗µβ∗µ + cβ4]gd4x

as a vacuum action. Now β∗µβ∗µ = (βµ + βκµ)(βµ + βκµ) and using (D.42) oneobtains

(D.44) −β2∗R + kβ∗µβ∗µ = −β2R + kβµβµ + (k − 6)β2κµκµ+

+6(β2κµ):µ + (2k − 12)βκµβµ

The term involving (β2κµ):µ can be discarded since its contribution to the actiondensity is a perfect differential, namely (β2κµ):µg = (β2κµg),µ and for the simplestvacuum equations one chooses k = 6 so that (D.43) becomes

(D.45) I =∫

[(1/4)FµνFµν − β2R + 6βµβµ + cβ4]gd4x

Thus I no longer involves the κµ directly but only via Fµν and I is invariant undertransformations κµ → κµ + φ,µ so the equations of motion that follow from theaction principle will be unaffected by such transformations (i.e. they have nophysical significance). Now consider three kinds of transformation:

(1) Any transformation of coordinates.(2) Any transformation of the metric gauge combined with the appropriate

transformation of potentials κµ → κµ + φ,µ.(3) In the vacuum one may make a transformation of potentials as above

without changing the metric gauge or alternatively one may transformthe metric gauge without changing the potentials. This works only wherethere is no matter.

For the field equations one makes small variations in all the field quantities gµν , κµ,and β, calculates the change in I and sets it equal to zero. Thus write

(D.46) δI =∫

[(1/2)Pµνδgµν + Qµδκµ + Sδβ)gd4x

and drop the cβ4g term since it is probably only of interest for cosmological pur-poses. One has

(D.47) δ[(1/4)FµνFµνg] = (1/2)Eµν

gδgµν − Jµgδκµ

402 APPENDIX D

with neglect of a perfect differential. Here Eµν is the EM stress tensor

(D.48) Eµν = (1/4)gµνFαβFαβ − FµαF να

and Jµ is the charge current vector

(D.49) Fµ = Fµν:ν = g

−1(Fµνg),ν

Considerable calculation and neglect of perfect differentials leads finally to

(D.50)Pµν = Eµν + β2[2Rµν − gµνR]− 4gµνββρ

:ρ + 4ββµ:ν + 2gµνβσβσ − 8βµβν ;

Qµ = −Jµ; S = −2βR− 12βµ:µ

and the field equations for the vacuum are

(D.51) Pµν = 0, Qµ = 0; S = 0

These are not all independent since

(D.52) P σσ = −2β2R− 12ββσ

:σ = βS

so the S equation is a consequence of the P equations. If one omits the EM termfrom the action it becomes the same as the Brans-Dicke action except that thelatter allows an arbitrary value for k; with k = 6 the vacuum equations are inde-pendent so the BD theory has one more vacuum field equation, namely (β2) = 0.

Now the action integral is invariant under transformations of the coordinatesysem and transformations of gauge; each of these leads to a conservation lawconnecting the quantities Pµν , Qµ, S defined via (D.46). For coordinate transfor-mations xµ → xµ + bµ one gets

(D.53)−δgµν = gµσbσ

,ν + gνσbσ,µ + gµν,σbσ; −δβ = βσbσ; −δκµ = κσbσ

,µ + κµ,σbσ

Putting these variations in (D.46) yields

(D.54) δI = −∫

[(1/2)Pµν(gµσbσ,ν + gνσbσ

,µ + gµν,σbσ)+

+Qµ(κσbσ,µ + κµ,σbσ) + Sβσbσ]gd4x =

=∫

[(Pµσ g),µ − (1/2)Pµνgµν,σg + (Qµκσg),µ −Qµκµ,σg− Sβσg]bσd4x

This δI vanishes for arbitrary bσ so one puts the coefficient of bσ equal to zero;using

(D.55) (Pµσ g),µ − (1/2)Pµνgµν,σg = Pµ

σ:µg; (Qµκσg),µ = κσQµ:µg + κσ,µQµ

g

this reduces to

(D.56) Pµσ:µ + κσQµ

:µ + FσµQµ − Sβσ = 0

Next consider a small transformation in gauge

(D.57) δgµν = 2λgµν , δβ = −λβ; δκµ = [log(1 + λ)],µ = λµ


Putting this in (D.46) yields

(D.58) δI =∫

[Pµνλgµν + Qµλµ − Sλβ]gd4x =∫

[Pµµ g− (Qµ

g),µ − Sβg]λd4x

Putting the coefficient of λ equal to zero gives Pµµ − Qµ

:µ − Sβ = 0 which with(D.56) comprise the conservation laws. For the vacuum one sees that (D.58) is thesame as (D.52) since Qµ

:µ = 0 from (D.50); also (D.56) reduces to

(D.59) Pµσ:µ + FσµQµ − β−1βσPµ

µ = 0

which may be considered as a generalization of the Bianchi identities. The conser-vation laws (D.56) and (D.58) hold more generally than for the vacuum, namelywhenever the action integral can be constructed from the field variables gµν , κµ, βalone.

Now let the coordinates of a particle be zµ, functions of the proper time smeasured along its world line. Put dzµ/ds = vµ for velocity so vµvµ = 1 and vµ

is a co-vector of power -1. One adds to the action the further terms

(D.60) I1 = −m

∫βds; I2 = e

∫β−1β∗µvµds

(m and e being constants). Then these terms are in-invariants with

(D.61) I2 = e

∫(β−1βµ + κµ)vµds = e

∫[(d/ds)(log(β)) + κµvµ]ds

and the first term contributes nothing to the action principle. Thus I2 = e∫

κµvµdswhich is unchanged when κµ → κµ+φ,µ since the extra term is e

∫(dφ/ds)ds. Thus

for a particle with action I1 + I2 the transformations (3) above are still possible.Now some calculation yields

(D.62) m[gµσd(βvµ)/ds + βΓσµνvµvν − βσ] =

= −evµFµσ ≡ m[d(βvµ)/ds + Γµρσvρvσ − βµ] = eFµνvν

This is the equation of motion for a particle of mass m and charge e; if e = 0it could be called an in-geodesic. If one works with the Einstein gauge then thecase e = 0 gives the usual geodesic equation. Next one considers the influence theof particle on the field and this is done by generating a dust of particles and acontinuous fluid leading to an equation

(D.63) ρ[(βvµ),ν + Γµασvαvσ − βµ] = σµν

where ρ and σ refer to mass and charge density respectively.

APPENDIX E

BICONFORMAL GEOMETRY

We sketch here some beautiful work of J. Wheeler [992, 994] which hasled to a number of important developments in mathematical physics (cf. also[35, 36, 987, 980, 990, 991, 993]). The paper [992] (not published) gives avery nice discussion of normal biconformal spaces and lays some of the foundationfor some later work of Wheeler et al for which [994] is apparently the best startingpoint. We will therefore extract here from [994] and remark that this work alonetranscends earlier approaches to unifying GR and EM. One works over an 8-Dbase space where in a flat situation the 4-D biconformal cospace to 4-D Minkowskispace corresponds to a standard tangent space (or momentum space) with vari-ables pµ transforming via L−1 under Lorentz transformations L of xµ (or dxµ).The symplectic form given by the exterior derivative of the Weyl 1-form providesa typical Hamiltonian dynamical structure and one gives general necessary andsufficient conditions for curved 8-D geometry to be in 1-1 correspondence with 4-D Einstein-Maxwell spacetime; further a consistent unified geometrical theory ofgravity and electromagnetism is obtained. This is very powerful stuff and we canonly sketch a few items here. Some connections of biconformal geometry to QMare given in Section 3.5.1 and we refer to [987, 994] for history and philosophy.

The conformal group is the most general set of transformations preservingratios of infinitesimal lengths. On a 4-D spacetime this group is 15 dimensional,with Lorentz transformations (6), translations (4), 4 inverse translations (specialconformal transformations), and dilations (1). One concentrates on flat situa-tions and develops biconformal structure as a conformal fiber bundle. This isconstructed as the quotient C/C0 of the conformal group by its isotropy subgroup(7-D homogeneous Weyl group) producing a conformal Cartan connection on an8-D manifold; this is then generalized to a curved 8-D manifold with the 7-D ho-mogeneous Weyl group as fiber by the addition of horizontal curvature 2-formsto the group structure equations. The resulting 8-D base manifold is called abiconformal space and the full 15-D fiber bundle the biconformal bundle. Wewill try to illustrate this via the equations. One uses the O(4, 2) representationof the conformal group for notation where (A,B, · · · ) = (0, 1, · · · , 5). Then theO(4.2) metric is ηab = diag(1, 1, 1,−1) (a, b = 1, · · · , 4) with η05 = η50 = 1 and allother components are zero. Introducing a connection 1-form ωA

B one has covariant

405

406 APPENDIX E

constancy via

(E.1) DηAB = dηAB − ηCBωCA − ηACωC

B = 0

The conformal connection may be broken into 4 independent Weyl invariant parts,the spin connection ωa

b , the solder form ωa0 , the co-solder form ω0

a, and the Weylvector ω0

0 where the spin connection satisfies ωab = −ηbcη

adωcd and the remaining

components of ωAB are related via

(E.2) ω50 = ω0

5 = 0; ω55 = −ω0

0 ; ωa5 = −ηabω0

b ; ω5a = −ηabω

b0

These constraints reduce the number of independent connection forms ωAB to the

required 15 and one can run A,B · · · from 0 to 4 (with 5 implicit). The structureconstants of the conformal Lie algebra now lead immediately to the Maurer-Cartan(MC) equations of the conformal group as dωA

B = ωCBωA

C (wedge product is ass-sumed) or written out

(E.3) dωab = ωc

bωac + ω0

bωa0 − ηbcη

adω0dωc

0;

dωa0 = ω0

0ωa0 + ωb

0ωab ; dω0

a = ω0aω0

0 + ωbaω0

b ; dω00 = ωg

0o0a

Note that d in (E.2) includes partial derivatives in all eight of the base spacedirections and when using coordinates one will write (xµ, yν) corresponding toindex positions on (ωb

0, ω0b ). Also ∂µφ = ∂φ/∂xµ while ∂µφ = ∂φ/∂yµ. The

generalization of (E.3) to a curved base space is obtained via

(E.4) dωab = ωc

bωac + ω0

bωa0 − ηbcη

adω0dωc

0 + Ωab = ωc

bωac + ∆da

bc ω0dωc

0 + Ωab ;

dωa0 = ω0

0ωa0 + ωb

0ωab + Ωa

0 ; dω0a = ω0

aω00 + ωb

aω0b + Ω0

a; dω00 = ωa

0ω0a + Ω0

0

One calls the four types of curvature Ωab , Ωa

0 , Ω0a, Ω0

0 the Riemann curvature,torsion, co-torsion, and dilational curvature respectively. Horizontality requireseach of the curvatures to take the form

(E.5) ΩAB = (1/2)ΩA

Bcdωc0ω

d0 + ΩAc

Bdωd0ω0

c + (1/2)ΩAcdB ω0

cω0d

The connection of a flat biconformal space is in the standard flat form whenwritten as

(E.6) ω00 = αa(x)dxa − yadxa ≡Wadxa; ωa

0 = dxa;

ω0a = dya − (αa,b + WaWb − (1/2)W 2ηab)dxb; ωa

b = (ηacηbd − δadδc

b)Wcdxd

Note that the Weyl vector Wa = αa(x)− ya depends on an arbitrary 4-vector αa

and also on the 4 coordinates ya; the presence of αa gives the generality requiredfor the EM vector potential while the ya keeps the dilational curvature zero. Ingeneral the dilational gauge vector of biconformal space is of the form

(E.7) ω00 = ω0

0µ(x, y)dxµ + ω0µ0 (x, y)dyµ

(i.e. an 8-D vector field depending on 8 independent variables) but constraining thebiconformal geometry to have vanishing curvatures forces ω0

0 = (αµ(x) − yµ)dxµ

which is precisely the form required to give the Lorentz force law. Then one provesin [994] that when the curvatures of biconformal space ΩA

B = 0 there exist globalcoordinates xa, ya) such that the connection takes the standard flat form.

Note in (E.6) if one holds the y coordinates fixed then equations 1,2,4 are

BICONFORMAL GEOMETRY 407

the connection forms for a 4-D Weyl spacetime with conformally flat metric ηab;the remaining equation 3 is then simply a 1-form constructed from the Weyl-Riccitensor. However the dilational curvature of a 4-D Weyl geometry is given by thecurl of the Weyl vector, equivalent here to the curl of the arbitrary αa, and viewedfrom the Weylian 4-D perspective the solution gives unphysical size change; it isonly with the inclusion of the additional momentum variables proportional to ya

that the dilational curvature can be seen to vanish. It is thus seen that the actualmotion of a particle in biconformal space is 8 dimensional and one can in factinterpret biconformal space, and therefore conformal gauge theory, as a general-ization of phase space (with symplectic structure). Indeed for a given Hamiltoniansystem one can specify a unique flat biconformal space by judicious choice of thesolder and co-solder forms (see [994] for details); in particular the extra 4 dimen-sions are identified with momenta and the integral of the Weyl vector is identifiedwith action. In order to make a full identification of a biconformal space with anEinstein-Maxwell spacetime one can proceed as follows. The idea is that the solderform should satisy the Einstein equations with arbitrary matter as source and thevector potential obtained via αa = q(φ,−Ai) = −qAa should satisfy the Maxwellfield equations with arbitrary EM currents. One assumes for example that thetorsion is zero (but not the cotorsion) leading to constraints Ω0

0 = 0 and Ωa0 = 0

and, setting ωa0 = ea, one wants a completion fa to the ea basis in which Ωa

bac = 0.Further one posits two field equations ∗d ∗ dω0

0 = J = Ja(x)ea and ω0a = Ta + · · ·

where Ta = −(1/2)(Tab − (1/3)ηabT )eb. This can all be achieved and leads to thegeneral identification stated above (cf. [994]).

Now to set the stage for [35] and connections to QM we gather the materialfrom the appendices to [35] (where the formulation is different). The conformalgroup generators include Lorentz transformations Ma

b = −MbaηacMcb , transla-

tions Pa, special conformal transformations Ka, and dilatations D satisfying thecommutation relations

(E.8) [Mab,M

cd] = −(δc

bMad + ηdfηacMf

b + ηbdηaeM c

e − δadM c

b);

[Mab, Pc] = −(ηcbη

adPd − δac Pb); [Ma

b,Kd] = −(δd

b δac − ηadηbc)Dc;

[Pa,Kb] = 2M ba − 2δb

aD; [D,Kb] = Kb; [D,Pa] = −Pa

(note dilatation corresponds to dilation and both terms seem to be popular). Theconformal Lie algebra has two independent involutive automorphisms; the first is

(E.9) σ1 : (Mab , Pa,Ka, D) → (Ma

b ,−Pa,−Ka, D)

and this identifies the invariant subgroup used as the isotropy subgroup in thebicomformal gauging. The second is

(E.10) σ2 : (Mab , Pa,Ka, D) → (Ma

b ,−ηabKb,−ηabPb,−D)

and may be chosen to be complex conjugation in order to define what are called σC

representations of the algebra. Thus if we assume the generators to be complex,σC representations have Pa and Ka as complex conjugates, while Ma

b is real andD pure imaginary. As an illustration note that while both so(3) and su(2) have

408 APPENDIX E

involutive automorphisms the existence of a σC representation singles out su(2).Thus while

(E.11) [Ji, Jj ] = εijkJk; [τi, τj ] = εijkτk

are both invariant under

(E.12) ρ : (J1, J2, J3) → (−J1, J2,−J3); (τ1, τ2, τ3) → (−τ1, τ2,−τ3)

where [Jj ]ik = εijk and τj = −(i/2)σj (σj being the Pauli matrices - cf. (B.25)) it isonly with the complex representation that ρ = ρC : (τ1, τ2, τ3) = (−τ1, τ2,−τ3) =ρ(τ1, τ2, τ3). As examples of conformal representations with this property firstconsider the covering group SU(2, 2), whose Lie algebra is isomorphic to thatof O(4, 2). Due to the local isomorphism between Spin(4, 2) and SU(2, 2) thisalgebra can be represented via spinors. Thus using 4 × 4 Dirac matrices γa onecan write su(2, 2) via

(E.13) γa, γb = 2ηab = 2diag(−1, 1, 1, 1) (a, b = 0, , 1, 2, 3)

One also defines

(E.14) σab = −(1/8)[γa, γb]; γ5 = iγ0γ1γ2γ3

where the full Clifford algebra has the basis

(E.15) Γ = 1, i1, γa, σab, iσab, γ5γa, iγ5γ

a, γ5, iγ5

The conformal Lie algebra may be obtained from this set be demanding invarianceof a spinor metric Q given by Q = iγ0. Then if one requires QΓ + Γ†Q = 0 thegenerators of the conformal Lie algebra are (cf. [35])(E.16)Ma

b = ηbcσac; Pa = (1/2)ηab(1 + γ5)γb; Ka = (1/2)(1− γ5)γa; D = −(1/2)γ5

Choosing any real representation for the Dirac matrices γ5 is necessarily imaginaryand it follows that

(E.17) Mab = Ma

b; Pa = ηabKb; D = −D

so the action of σC is realized. Alternatively one may consider a complex functionspace representation of the conformal algebra via

(E.18) Mµν = −1

2

(zµ ∂

∂zν+ zµ ∂

∂zν− zν

∂

∂zµ− zν ∂

∂zµ

);

D = zµ ∂

∂zµ− zν ∂

∂zν; Pµ =

∂

∂zµ+(

zµzν − 12z2δν

µ

)∂

∂zν;

Kµ =∂

∂zµ+(

zµzν − 12z2δν

µ

)∂

∂zν

Note the generators are complex but the group manifold is real. In either of theserepresentations the Maurer-Cartan (MC) equations inherit the same symmetryunder σC and in particular the gauge vector of dilatations (the Weyl vector) isimaginary. To clarify this one shows that the dilatations generated by an imaginary

BICONFORMAL GEOMETRY 409

D nevertheless give a real factor as expected. Thus first consider su(2, 2) with abasis of Dirac matrices in which

(E.19) D = −12γ5 = −1

2

(−σy

σy

); σy =

(−i

i

)Define the definite conformal weight spinors χA, ψB via

(E.20) Dχ = (1/2)χ; Dψ = −(1/2)ψ; eλDχ = eλ/2χ; eλDψ = e−λ/2ψ

For the complex function space representation of the conformal group (E.18) onehas (for one variable z = rexp(iφ))

(E.21) D ∼ −i∂

∂φ

so D measures the phase of a complex number. Homogeneous functions of z andz are then eigenfuncitons and D measures the degree of homogeneity. Thus iff(z, z) = zazb there results exp(λD)f(z, z) = exp[(a − b)λ]f(z, z) so there aredilatations with the weight of the function encoded into the total phase. Similarlyin multiple complex dimensions eigenfunctions can be built up from powers of thenorms

(E.22) fα−β = (√

z2)α(√

z2)β ; Dfα−β = D(z2)α/2(z2)β/2 = (α− β)fα−β

Note that zaza is of weight zero with D(zaza) = 0.

Gauge transformations will remain real even though there is a complex valuedconnection. A local gauge transformation is given via

(E.23) Λ = MabΛ

ba + DΛ0

Note Λ is complex, since Λba, Λ0 are real parameters used to exponentiate the

generators M (real) and D (imaginary), and it follows that a gauge transformationof the Weyl vector is δω = −dΛ0 where Λ0; one can then define a scale covariantderivative of a definite weight scalar field via

(E.24) Df = df + kωf

where k is the conformal weight of f . To see that this is a gauge invariant expres-sion one takes a dilatational gauge transformation

(E.25) f ′ = fexp(kΛ0); ω′ = ω + δω = ω − dΛ0

which implies

(E.26) D′f ′ = d(fexp(kΛ0)) + k(ω − dΩ0)f = exp(kΛ0)Df

Thus the equation is covariant and the MC structure equations are invariant un-der real scalings. Of course in generic gauges the Weyl vector is complex butthe invariance fo the structure equations under gauge transformations guaranteesconsistency. Note also that whether the Weyl vector is complex or pure imaginaryexp(

∮ω) remains a pure phase since the Weyl vector is pure imaginary in at least

one gauge and the above expression is gauge invariant. Finally one writes theCartan structure equations for flat σC biconformal space via

(E.27) dωab = ωc

bωac + 2ωbω

a; dωa = ωcωac + ωωa; dω = 2ωaωa

410 APPENDIX E

(and their conjugates); here ωa corresponds to translation generators, ω is theWeyl vector, and ωa

b is the spin connection. A first order perturbative solution is

(E.28) ωab (δa

e ηcb − δac ηeb)xcdxe + (δa

e ηcb − δac ηeb)ycdye;

ω = i(yadxa − xadya); ωa = dxa + idya+

+(−1

2xaxe +

i

2(δa

e xcyc − xaye) +

12yaye

)(dxe − idye)

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-

Index

Abraham Lorenz equation, 224

absolute uncertainty, 77

acausality, 224

accretion, 156

action principle, 7

additional symmetries, 347

Arnowitt Deser Misner (ADM)decomposition, 161, 170, 181, 387

aether, 212

equations, 214, 218

field equation, 219

reference frame, 214

affine connection, 106

age of universe, 153

angular momentum, 43, 384

annihilation, 11, 229

antiparticles, 65

Ashtekhar variables, 157, 162, 390, 391

average action principle, 119

average kinetic energy, 117

average momentum potential, 7, 117

average stationary action principle, 121

backward derivative, 28

backward mean value, 15

beable, 185, 189

Beck Cohen entropy, 256

Bell’s theorem, 61

Bell type quantum field theory, 78

Berry phase, 292

Bessel function, 183

Bianchi identity, 102, 114, 403

bias, 261

biconformal coordinates, 141

cospace, 405

gauging, 407

geometry, 405

manifold, 139

space, 137

bispinor, 210

bivelocity, 14

blackbody gamma distribution, 24

radiation, 204

spectrum, 205

Bogoliubov transformation, 59

Bohmian Einstein equations, 158

Hamilton Jacobi equation, 158

mechanics, 5, 40

noncommutative quantum cosmology,

185

quaantum field theory, 280

quantum geometry, 102

quantum gravity, 188

quantum mass field, 282

trajectory, 187

Bohr radius, 156

Boltzman law, 239

constant 241, 248, 271

entropy, 247

Boltzman Gibbs distribution, 257

entropy, 259

Boolean algebra, 335

Borel subalgebra, 354

Borgnis function, 314

Bose Einstein condensate, 307

boson, 331

bound information, 268

box dimension, 14

Braffort Marshall equation, 205

Brans-Dicke theory, 104, 395

action, 402

Brownian motion, 1, 11

noise, 82

trail, 14

canonical general relativity, 109

transformation, 196 item canonicallyconjugate, 195

Cantorian space, 23

Cantor set, 23, 33

Caratheodory complete figure, 121

canonical theory, 372

437

438 INDEX

Cartan structure equations, 409

Casimir effect, 203, 205

operator, 354

Cauchy data, 112

hypersurface, 56, 58

problem, 380

Littlewood formula, 344

causality, 191, 198

cavitation, 293

central extension, 342, 347

chaotic Hamiltonian, 252

orbit, 253

channels, 69

Chapman Kolmogorov equation, 81

Charlier polynomial, 318

Chern Simons (CS) theory, 393

Christoffel symbols, 91, 105, 144, 379, 387

circle product, 320, 321

classical Hamilton Jacobi equation, 49

statistical mechanics, 120

kinetics, 276

classically chaotic Hamiltonian, 252

Clebsch-Gordon coefficients, 354

Clifford algebra, 341, 357

group, 350

clock, 164

cloud, 40, 97

coassociative, 319

cocommutative, 319

cocycle condition, 11, 51

combinatorics, 319, 356

complete integral, 375

complex covariant derivative, 129

derivative, 28

diffusion coefficient, 21

energy, 19

momentum, 129

potential, 20

time derivative, 19

velocity, 12, 15, 19, 285

Compton time, 153

wave length, 57, 127, 152

configuration information entropy, 8

conformal factor, 96, 99

fibre bundle, 405

gauge theory, 138

general relativity, 111

geometry, 104

group, 138

Lie algebra, 407

transformation, 96, 405

weight, 409

congruence of curves, 121

coherent state, 250

connection, 405

coefficients, 91

one form, 405

Connes Moscovici algebra, 340

conserved current, 41

consistent formulation of gravity, 108, 110,

286

constraint algebra, 158, 173

continuity equations, 6, 9, 51, 54, 80, 94,

117, 162, 197, 20, 299

Copenhagen, 172

coproduct, 319, 331

coquasitriangular structure, 327

Coriolis force, 302

cosmic matter, 148

cosmological constant, 99, 127,, 144

cosolder form, 406

Coulomb gauge, 47, 216, 222

self interaction, 223

counit, 329

coupling identity, 322

covariance matrix, 198, 200

covariant conservation law, 57

derivative, 46, 56, 114

Cramer-Rao inequality, 6, 117, 245

creation, 11, 65, 229, 277, 326

of particles, 153

cross entropy, 118

current, 5, 60, 64

curvature tensor, 93, 144

Darboux theorem, 140

dark matter, 61, 151, 156

Davies Unruh effect, 205

deBroglie Bohm (dBB), 63

deBroglie wave length, 127, 151

decoherence, 172

deDonder Weyl canonical form, 373

formalism, 71

density, 5

operator, 240

matrix, 249

deWitt metric, 181, 287

diffeomorphism constraint, 163, 167

differential entropy, 239, 244

diffusion current velocity, 10

process, 12

dilatation (dilation), 138, 405

dilatational curvature, 138, 406

gauge transformation, 409

generator, 139

operator, 126

dilaton, 111

Dirac aether, 207

algebra, 389

equation, 47, 383

INDEX 439

charge, 150

dual tensor, 380

gauge function, 145

quantization, 292

Weyl action, 289, 397

Weyl geometry, 290

dissipative device, 37

distribution (Schwartz), 34

Doppler shift, 209, 228

double slit experiment, 78

dust dominated era, 148

Dynkin diagram, 352

effective field theory, 201

Hamiltonian, 254

mass, 64

effectivity, 64, 68

Ehrenfest equations, 3, 253

Einstein aether, 307

field equations, 201, 282, 287, 311

frame, 104

gauge, 145

Hilbert action, 388, 391

Maxwell spacetime, 407

tensor, 98, 145, 387

electromagnetic (EM) tensor, 45, 60, 97

field strength, 144

emergence, 152

emergent symmetry, 77

energy bounded below, 41

entropy balance, 256

momentum, 20

ensemble, 97, 191

average, 124

entropy, 119, 278

balance, 9

equipartition, 257

equivariance, 67, 80, 288

equivalence principle, 48, 50, 52, 55, 278

equivariant jump rates, 85

ergodic clump, 40

escort density operator, 256

probability, 257

Euler angles, 301, 305

equation, 2, 11, 276

Lagrange equations, 21, 296

Eulerian density, 299

velocity, 299

exact uncertainty, 7, 190, 289

extreme physical information (EPI), 263,265

extremal motion, 138

extrinsic curvature, 157, 172

F continuous, 33

integrability, 32, 33

Faa da Bruno algebra, 339

factor ordering, 173

Feynman path integral, 136

propagator, 322, 328

quantum rules, 120

Fick law, 10

field redefinition, 221

theory, 20

finite particle vector, 328

Fisher Euler theorem, 270,

information, 10, 151, 238, 240, 242, 247,24, 276, 285, 293

information matrix, 8, 118

information measure, 241, 248

length, 117

metric, 190, 245

temperature, 263

time, 262

flat biconformal space, 406

Floydian time, 5, 40, 49

flux density, 29

Fock space, 229, 341, 348

Fokker-Planck equation, 9, 15, 136

foliation, 74

forward derivative, 28

mean velocity, 15

fractal dimension, 15, 126

geometry, 126

path, 25

spacetime, 22, 130

spray, 26

string, 26

fractional derivative, 30

free energy, 50

fall, 130

free bialgebra, 338

coproduct, 336

fermions, 341, 348

Friedman Robertson Walker (FRW) lineelement, 148

universe, 185, 288

Frobenius integrability condition, 366

Fubini Study metric, 231, 237

fuzzy, 37, 40

gamma matrices, 48

dependence, 134

freedom, 135

transformation, 93

vector, 115

gauge constraint, 163

Gauss constraint, 162

Codazzi equations, 387

law, 223

Gaussian, 237, 249

440 INDEX

generalized Euler equations, 100

equivalence principle, 101

Legendre transformation, 369

momentum, 135

Schrodinger equation, 22

tau function, 342

thermodynamic potential, 270

generating function, 358

geodesic equation, 16, 126, 155

geometric phase, 291

Gibbs canonical distribution, 247

ensemble, 120

golden ratio, 23

graded bialgebra, 338

gradient condition, 113

grading, 336

Grassman fields, 66

number, 65

manifold, 350

gravitational mass, 205

gravity quantum interaction, 100

guidance equation, 76, 161, 174

Hamiltonian constraint, 162, 167

vector field, 234

Hamilton Jacobi equation (HJE), 62, · · ·Hamilton’s principal function, 49, 140

Hartree equation, 156

Hausdorff dimension, 14, 23, 25

Hawking radiation, 59

heat kernel regularization, 167

Heisenberg algebra, 342, 345, 363

equation, 48

picture, 62

helicity 228

equation, 211

state, 229

Helmholz conservation law, 242

free energy, 243, 268

Hermite function, 62

Hermitian metric, 233

hidden variables, 79, 315

Higgs field, 149, 206

Hirota bilinear identity, 342, 351

formulas, 319

Miwa variables, 343

Hodge star operator, 45, 387

Hopf algebra, 320, 339

horizontal curvature, 401

Hubble constant, 149, 153

Hurst exponent, 26

hydrodynamics, 2, 276, 293

hydrodynamical model, 28

hydrostatic potential, 17

pressure, 18

hypergeometric series, 345

inertial mass, 101, 152, 205

information, 118

demon, 265

entropy, 10, 243, 253

independence, 191, 198

in-geodesic, 403

invariant, 143

tensor, 398

integrable spacetime, 144

structures, 234

systems, 353

Weyl connection, 120

Weyl Dirac geometry, 143, 148, 289

integral curves, 71

manifold, 367, 371

interacting quantum fields, 331

internal energy, 296

Hodge dual, 392

stress tensor, 29

angular momentum, 305

intertwining, 349

intrinsic curvature, 172

information, 266

invariance, 191, 198

irrotational, 297

Jack polynomial, 342

Jackson derivative, 261

Jacobian, 50, 51, 125, 303

joint EM density, 145

probability density, 112

Jordan frame, 104

product, 232, 236

jump, 79, 83

KP hierarchy and theory, 342, 351

Kahler function, 233

geometry, 286

isomorphism, 235

manifold, 233

space, 231

Kaluza Klein, 132

Kaniadakis entropy, 256

Kantowski Sachs universe, 181, 182

Kepler potential, 156

kinematic pressure, 17

Klein Gordon equation, 41, · · ·Kolgomorov representation, 82

Sinai entropy, 253

Kullback entropy, 241

Leibler entropy, 264

Lagrangian flow equations, 304

Laguerre polynomial, 156

INDEX 441

Landsberg, Vedral, Rajagopal, Abeentropy, 261

Langevin equation, 224

Laplace Hopf algebra, 320

pairing, 330

rules, 120

lapse function, 157, 166, 285, 287

large numbers, 398

Lax operator, 50

Legendre duality, 49

transformation, 268, 271, 308, 367

Leibnitz relation, 324

Lepage method, 367

equivalence relation, 368

Levi Civita connection, 231, 294, 387

tensor, 136

Levy Leblond phase, 292

Lichnerowicz York metric, 162

Lie derivative, 366

LieWehrl bound, 252

likelihood function, 246

linear SED, 221

linearized aether stress tensor, 219

Einstein tensor, 219

London equation, 137

loop quantum gravity, 393

Lorenz boost, 43

force, 204, 402

gauge, 220

group, 43

invariance, 42, 46, 94

metric, 379

symmetry, 218

transformation, 138

violation, 218, 220

Lyapunov coefficient, 262

exponent, 255

spectrum, 252

Macdonald function, 355

Mach’s principle, 153

macrogravitational coupling constant, 155

Madelung, 2, 4, 112

manifest covariance, 88

many fingered time, 87

worlds, 172

Markov chain, 81

process, 81

wave equation, 11

mass function, 31

generator, 52

of universe, 153

massless Dirac equation, 211

KG equation, 307

particle, 210

matrix model, 357

matter creation, 147

Maurer Cartan equations, 406, 408

maximum entropy, 245

Maxwell equations, 45, 93,· · ·metric redefinition, 221

tensor, 283

microstate, 52, 315

minimal free generator, 86

minimum Fisher information, 269

minisuperspace, 164, 174,41, 186, 188

Minkowski Bouligand dimension, 23

length, 139

Misner parametrization, 182

Mobius transformation, 49

modified Einstein equations, 97, 161, 282,286

momentum fluctuation, 192

potential, 117, 197

Moyal bracket, 184

product, 288

Navier Stokes equations, 17, 29

neutrino, 218

Newton constant, 171

equation, 28

law, 153

Nijenhuis tensor, 231

Noether current, 213

theorem, 4

noncommutative geometry, 151

Schrodinger equation, 178

nonequilibrium system, 251

nondifferentiable, 16, 27, 278

nonlinear Schrodinger equation (NLSE), 4,18, 20, 245

normal biconformal space, 405

ordering, 42, 56

product, 321, 326, 329

normalizable, 43

normalization, 44

observable, 61

operator ordering ambiguity, 165

product, 320, 357

orbitals, 156

osmotic velocity field, 10, 240, 243, 277,279

Palatini action, 390, 391

parallel transport, 46, 105

particle current, 57, 60

partition function, 251, 357

path space, 121

integral, 134

Pauli matrices, 408

442 INDEX

perfect, 31, 34

Pesin theorem, 262

Pfaffian form, 365

phase space distribution, 250

transition, 247

photon, 227

equation, 210, 312

tensor, 227

physical metric, 109

pion, 153

pilot equation 76

wave cosmology, 169

Pinter Hopf algebra, 340

identity, 325

Planck constant, 194, 225, 226, 236, 246

length, 40, 127

scale, 153

Planckian constant, 194

cosmic egg, 148

Poincare invariant, 94

lemma, 365

Lie algebra, 138

Poisson bracket, 155, 158

polarization, 212

positive operator valued measure, 85

power absorbtion, 242

law, 126, 255

release, 242

removal, 242

Poynting vector, 204, 301

preferred reference frame, 220

prematter, 147

prePlanckian period, 148

prepotential, 50, 308

pressure, 2, 296

probability density, 6

projection valued measure, 79, 85

projective Hilbert space, 233

pseudodifferential operator, 347

psi (ψ) aether, 307, 312, 313

pure states, 235

quantization, 8, 119

quantum action, 256

chaos, 255

cosmology, 171, 188, 286

effects of matter, 96

Einstein equations, 160, 285

Einstein Hamilton Jacobi equation, 162

equilibrium, 77

equilibrium condition, 170

equilibrium density, 6, 41

equilibrium distribution, 77

equilibrium hypothesis, 6, 41, 77

expectation value, 3

fields, 43

field theory, 42, · · ·fluctuations, 66, 151

fluid, 4

force, 45, 111, 282, 283, 293

fractal, 34, 36

general relativity, 163

geometrodynamics, 176

geometry, 111, 285

gravity, 157, 170

group, 319, 327

Hamilton Jacobi equation (QHJE), 100,308

Lagrangian, 315

mass, 102, 103, 278

measurement, 132

path, 14

potential, 2, 17, 29, 39,25, 70, 73, 76, 89,

94, 6, 98, 99, 100, 111, 116, 117, 133,

151, 158, 159, 162, 21, 166, 170, 172,176, 177, 190, 238, 254, 255, 275, 277,8, 293, 295, 298, 300, 306, 308, 310,

311

stationary HJE (QSHJE), 48

stress tensor, 293

trajectory, 315, 317

vacuum, 176

quasitriangular Hopf algebra, 327

quintessence, 151

radiation dominated universe, 147

radiative reactive forces, 11

random curve, 112

momentum fluctuations, 6

motion, 112

walk, 14

rays, 235, 238

refinement relation, 335

regularization, 164

relativistic classical HJE (RCHJE), 50

covariant derivative, 155

Gibbs ensemble, 123

quantum mass, 95

quantum potential, 279

renormalization, 126, 322

algebra, 337

coproduct, 334

renormalized T maps, 325

t product, 325

Renyi entropy, 256, 261

Wooters information, 264

rest frame, 218

Ricatti equation, 270

Ricci scalar, 157, 159, 218, 284

scalar curvature, 99, 294

INDEX 443

tensor, 93, 380

Weyl curvature, 152

Riemannian curvature, 406

flat space, 107

geometry, 92

Riemann bracket, 232, 236

Christoffel tensor, 410

Silberstein vector, 300

tensor, 46, 379

Rolle’s theorem, 33

scalar curvature, 113, 221, 294, 311

tensor theory, 282

scale laws, 126

relativity, 14, 278

scaling dimensions, 247

factor, 310

Schouten Haantjes conformal mass, 133

Schrodinger cat, 3

equation, 1, · · ·operators, 44

picture, 62, 66

representation, 171

Schur function, 343

Schwarzian connection, 49

derivative, 49

Schwarzschild black hole, 153

surface, 143

Schwinger deWitt function, 59

second quantization, 41

self dual action, 391

sensitivity matrix, 252

Shannon entropy, 239

information, 249, 272

shift functions, 157, 285, 289

singularity, 175

skew Schur function, 348

slope function, 374

smearing, 43

Smoluchowski diffusion, 11, 242

smoothed fields, 329

solar mass, 154

solder form, 406

source, 67

equations, 397

spacetime foliation, 294

special conformal transformation, 138, 405

spectral zeta function, 26

specific heat, 261

spin, 48, 230, 301, 384

covector, 406

spinor, 65

line orbital motion, 208

metric, 408

wavefunction, 210

staircase function, 31

standing waves, 314

star structure, 329

statistical determination of geometry, 283

transparency, 121

stochastic electrodynamics (SED), 221

jump, 80

Newton equation, 13

quantization, 13

quantum mechanics, 27

Stokes theorem, 366

stress energy tensor, 106

strong Markov property, 81

subquantum statistical ensemble, 4, 276

superconducting, 137

superluminal, 62, 306

Sweedler’s crossed product, 327

symmetric Hopf algebra, 327

Fock space, 327

symplectic manifold, 234

tachyons, 94

t-map, 324

T map, 324

Takabayashi stress tensor, 295

tau function, 341, 348, 351

tension, 147

thermal energy, 204

equilibrium, 251

fluctuations, 251

stability, 258

thermodynamic equilibrium, 261

entropy, 258

force, 10, 242

time evolution operator, 66

ordering, 319

ordered product, 330

Toda lattice, 342

Tomonaga Schwinger equation, 87

topological dimension, 25

torsion, 133, 402

transformation of units, 108

transition amplitude, 256

functions, 82

scale, 154

translations, 138

transmutation, 354

transplantation, 92, 282, 396

transversality, 227

Tsallis entropy, 256, 259

twisted product, 327, 330

two point function, 59

uncertainty, 53, 98, 141, 178, 203

universal enveloping algebra, 352

upper box dimension, 23

444 INDEX

vacuum, 42, 65

energy, 61, 208, 350

expectation, 230

state, 229

state, 342

van der Waals binding, 205

vector constraint, 162

velocity boost, 40

Verma module, 352

Virasoro algebra, 50

virial theorem, 156

viscosity, 19

vonNeumann continuous geometry, 24

entropy, 239, 249, 251

vortices, 255

vortex lines, 298

wave function, 1, 3

weak solution, 36

Weierstrass function, 35

WKB, 50

Weyl algebra, 138

covariant derivative, 94

Dirac action, 101

Dirac theory, 143

field, 116, 285

gauge transformation, 143

geometry, 92, 109, 113, 284, 395

manifold, 110

measurement, 132

one form, 405

power, 143

quantization, 184

scalar charge, 150, 290

Ricci scalar curvature, 117

Ricci tensor, 407

scalar charge, 145

structure, 120

symbol, 177, 184

transformation, 132

vector, 102, 133, 138

weight, 136

Wheeler deWitt (WDW) equation, 70,157, 173, 183, 201

metric, 167

Wheeler supermetric, 171

Wiener Brownian motion, 14

integral, 141

process, 15

Whitaker function, 354

Wick’s theorem, 321,10, 351

Wigner function, 250

Wightman function, 55, 56

world lines, 78

(x,ψ) duality, 309, 311

Young diagram, 344

Z pairing, 322zero point field (ZPF), 42, 203Zitterbewegung, 203, 204, 207, 209


Series Editor: Alwyn van der Merwe, University of Denver, USA

1. M. Sachs: General Relativity and Matter. A Spinor Field Theory from Fermis to Light-Years.With a Foreword by C. Kilmister. 1982 ISBN 90-277-1381-2

2. G.H. Duffey: A Development of Quantum Mechanics. Based on Symmetry Considerations.1985 ISBN 90-277-1587-4

3. S. Diner, D. Fargue, G. Lochak and F. Selleri (eds.): The Wave-Particle Dualism. A Tribute toLouis de Broglie on his 90th Birthday. 1984 ISBN 90-277-1664-1

4. E. Prugovecki: Stochastic Quantum Mechanics and Quantum Spacetime. A Consistent Unifi-cation of Relativity and Quantum Theory based on Stochastic Spaces. 1984; 2nd printing 1986

ISBN 90-277-1617-X5. D. Hestenes and G. Sobczyk: Clifford Algebra to Geometric Calculus. A Unified Language

for Mathematics and Physics. 1984 ISBN 90-277-1673-0; Pb (1987) 90-277-2561-66. P. Exner: Open Quantum Systems and Feynman Integrals. 1985 ISBN 90-277-1678-17. L. Mayants: The Enigma of Probability and Physics. 1984 ISBN 90-277-1674-98. E. Tocaci: Relativistic Mechanics, Time and Inertia. Translated from Romanian. Edited and

with a Foreword by C.W. Kilmister. 1985 ISBN 90-277-1769-99. B. Bertotti, F. de Felice and A. Pascolini (eds.): General Relativity and Gravitation. Proceedings

of the 10th International Conference (Padova, Italy, 1983). 1984 ISBN 90-277-1819-910. G. Tarozzi and A. van der Merwe (eds.): Open Questions in Quantum Physics. 1985

ISBN 90-277-1853-911. J.V. Narlikar and T. Padmanabhan: Gravity, Gauge Theories and Quantum Cosmology. 1986

ISBN 90-277-1948-912. G.S. Asanov: Finsler Geometry, Relativity and Gauge Theories. 1985 ISBN 90-277-1960-813. K. Namsrai: Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics. 1986

ISBN 90-277-2001-014. C. Ray Smith and W.T. Grandy, Jr. (eds.): Maximum-Entropy and Bayesian Methods in Inverse

Problems. Proceedings of the 1st and 2nd International Workshop (Laramie, Wyoming, USA).1985 ISBN 90-277-2074-6

15. D. Hestenes: New Foundations for Classical Mechanics. 1986 ISBN 90-277-2090-8;Pb (1987) 90-277-2526-8

16. S.J. Prokhovnik: Light in Einstein’s Universe. The Role of Energy in Cosmology and Relativity.1985 ISBN 90-277-2093-2

17. Y.S. Kim and M.E. Noz: Theory and Applications of the Poincare Group. 1986ISBN 90-277-2141-6

18. M. Sachs: Quantum Mechanics from General Relativity. An Approximation for a Theory ofInertia. 1986 ISBN 90-277-2247-1

19. W.T. Grandy, Jr.: Foundations of Statistical Mechanics. Vol. I: Equilibrium Theory. 1987ISBN 90-277-2489-X

20. H.-H von Borzeszkowski and H.-J. Treder: The Meaning of Quantum Gravity. 1988ISBN 90-277-2518-7

21. C. Ray Smith and G.J. Erickson (eds.): Maximum-Entropy and Bayesian Spectral Analysisand Estimation Problems. Proceedings of the 3rd International Workshop (Laramie, Wyoming,USA, 1983). 1987 ISBN 90-277-2579-9

22. A.O. Barut and A. van der Merwe (eds.): Selected Scientific Papers of Alfred Lande. [1888-1975]. 1988 ISBN 90-277-2594-2


23. W.T. Grandy, Jr.: Foundations of Statistical Mechanics. Vol. II: Nonequilibrium Phenomena.1988 ISBN 90-277-2649-3

24. E.I. Bitsakis and C.A. Nicolaides (eds.): The Concept of Probability. Proceedings of the DelphiConference (Delphi, Greece, 1987). 1989 ISBN 90-277-2679-5

25. A. van der Merwe, F. Selleri and G. Tarozzi (eds.): Microphysical Reality and QuantumFormalism, Vol. 1. Proceedings of the International Conference (Urbino, Italy, 1985). 1988

ISBN 90-277-2683-326. A. van der Merwe, F. Selleri and G. Tarozzi (eds.): Microphysical Reality and Quantum

Formalism, Vol. 2. Proceedings of the International Conference (Urbino, Italy, 1985). 1988ISBN 90-277-2684-1

27. I.D. Novikov and V.P. Frolov: Physics of Black Holes. 1989 ISBN 90-277-2685-X28. G. Tarozzi and A. van der Merwe (eds.): The Nature of Quantum Paradoxes. Italian Studies in

the Foundations and Philosophy of Modern Physics. 1988 ISBN 90-277-2703-129. B.R. Iyer, N. Mukunda and C.V. Vishveshwara (eds.): Gravitation, Gauge Theories and the

Early Universe. 1989 ISBN 90-277-2710-430. H. Mark and L. Wood (eds.): Energy in Physics, War and Peace. A Festschrift celebrating

Edward Teller’s 80th Birthday. 1988 ISBN 90-277-2775-931. G.J. Erickson and C.R. Smith (eds.): Maximum-Entropy and Bayesian Methods in Science and

Engineering. Vol. I: Foundations. 1988 ISBN 90-277-2793-732. G.J. Erickson and C.R. Smith (eds.): Maximum-Entropy and Bayesian Methods in Science and

Engineering. Vol. II: Applications. 1988 ISBN 90-277-2794-533. M.E. Noz and Y.S. Kim (eds.): Special Relativity and Quantum Theory. A Collection of Papers

on the Poincare Group. 1988 ISBN 90-277-2799-634. I.Yu. Kobzarev and Yu.I. Manin: Elementary Particles. Mathematics, Physics and Philosophy.

1989 ISBN 0-7923-0098-X35. F. Selleri: Quantum Paradoxes and Physical Reality. 1990 ISBN 0-7923-0253-236. J. Skilling (ed.): Maximum-Entropy and Bayesian Methods. Proceedings of the 8th International

Workshop (Cambridge, UK, 1988). 1989 ISBN 0-7923-0224-937. M. Kafatos (ed.): Bell’s Theorem, Quantum Theory and Conceptions of the Universe. 1989

ISBN 0-7923-0496-938. Yu.A. Izyumov and V.N. Syromyatnikov: Phase Transitions and Crystal Symmetry. 1990

ISBN 0-7923-0542-639. P.F. Fougere (ed.): Maximum-Entropy and Bayesian Methods. Proceedings of the 9th Interna-

tional Workshop (Dartmouth, Massachusetts, USA, 1989). 1990 ISBN 0-7923-0928-640. L. de Broglie: Heisenberg’s Uncertainties and the Probabilistic Interpretation of Wave Mechan-

ics. With Critical Notes of the Author. 1990 ISBN 0-7923-0929-441. W.T. Grandy, Jr.: Relativistic Quantum Mechanics of Leptons and Fields. 1991

ISBN 0-7923-1049-742. Yu.L. Klimontovich: Turbulent Motion and the Structure of Chaos. A New Approach to the

Statistical Theory of Open Systems. 1991 ISBN 0-7923-1114-043. W.T. Grandy, Jr. and L.H. Schick (eds.): Maximum-Entropy and Bayesian Methods. Proceed-

ings of the 10th International Workshop (Laramie, Wyoming, USA, 1990). 1991ISBN 0-7923-1140-X

44. P. Ptak and S. Pulmannova: Orthomodular Structures as Quantum Logics. Intrinsic Properties,State Space and Probabilistic Topics. 1991 ISBN 0-7923-1207-4

45. D. Hestenes and A. Weingartshofer (eds.): The Electron. New Theory and Experiment. 1991ISBN 0-7923-1356-9


46. P.P.J.M. Schram: Kinetic Theory of Gases and Plasmas. 1991 ISBN 0-7923-1392-547. A. Micali, R. Boudet and J. Helmstetter (eds.): Clifford Algebras and their Applications in

Mathematical Physics. 1992 ISBN 0-7923-1623-148. E. Prugovecki: Quantum Geometry. A Framework for Quantum General Relativity. 1992

ISBN 0-7923-1640-149. M.H. Mac Gregor: The Enigmatic Electron. 1992 ISBN 0-7923-1982-650. C.R. Smith, G.J. Erickson and P.O. Neudorfer (eds.): Maximum Entropy and Bayesian Methods.

Proceedings of the 11th International Workshop (Seattle, 1991). 1993 ISBN 0-7923-2031-X51. D.J. Hoekzema: The Quantum Labyrinth. 1993 ISBN 0-7923-2066-252. Z. Oziewicz, B. Jancewicz and A. Borowiec (eds.): Spinors, Twistors, Clifford Algebras and

Quantum Deformations. Proceedings of the Second Max Born Symposium (Wrocław, Poland,1992). 1993 ISBN 0-7923-2251-7

53. A. Mohammad-Djafari and G. Demoment (eds.): Maximum Entropy and Bayesian Methods.Proceedings of the 12th International Workshop (Paris, France, 1992). 1993

ISBN 0-7923-2280-054. M. Riesz: Clifford Numbers and Spinors with Riesz’ Private Lectures to E. Folke Bolinder and

a Historical Review by Pertti Lounesto. E.F. Bolinder and P. Lounesto (eds.). 1993ISBN 0-7923-2299-1

55. F. Brackx, R. Delanghe and H. Serras (eds.): Clifford Algebras and their Applications inMathematical Physics. Proceedings of the Third Conference (Deinze, 1993) 1993

ISBN 0-7923-2347-556. J.R. Fanchi: Parametrized Relativistic Quantum Theory. 1993 ISBN 0-7923-2376-957. A. Peres: Quantum Theory: Concepts and Methods. 1993 ISBN 0-7923-2549-458. P.L. Antonelli, R.S. Ingarden and M. Matsumoto: The Theory of Sprays and Finsler Spaces

with Applications in Physics and Biology. 1993 ISBN 0-7923-2577-X59. R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applications.

1994 ISBN 0-7923-2591-560. G. Adomian: Solving Frontier Problems of Physics: The Decomposition Method. 1994

ISBN 0-7923-2644-X61. B.S. Kerner and V.V. Osipov: Autosolitons. A New Approach to Problems of Self-Organization

and Turbulence. 1994 ISBN 0-7923-2816-762. G.R. Heidbreder (ed.): Maximum Entropy and Bayesian Methods. Proceedings of the 13th

International Workshop (Santa Barbara, USA, 1993) 1996 ISBN 0-7923-2851-563. J. Perina, Z. Hradil and B. Jurco: Quantum Optics and Fundamentals of Physics. 1994

ISBN 0-7923-3000-564. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 1: The Field B(3). 1994

ISBN 0-7923-3049-865. C.K. Raju: Time: Towards a Constistent Theory. 1994 ISBN 0-7923-3103-666. A.K.T. Assis: Weber’s Electrodynamics. 1994 ISBN 0-7923-3137-067. Yu. L. Klimontovich: Statistical Theory of Open Systems. Volume 1: A Unified Approach to

Kinetic Description of Processes in Active Systems. 1995 ISBN 0-7923-3199-0;Pb: ISBN 0-7923-3242-3

68. M. Evans and J.-P. Vigier: The Enigmatic Photon. Volume 2: Non-Abelian Electrodynamics.1995 ISBN 0-7923-3288-1

69. G. Esposito: Complex General Relativity. 1995 ISBN 0-7923-3340-3


70. J. Skilling and S. Sibisi (eds.): Maximum Entropy and Bayesian Methods. Proceedings of theFourteenth International Workshop on Maximum Entropy and Bayesian Methods. 1996

ISBN 0-7923-3452-371. C. Garola and A. Rossi (eds.): The Foundations of Quantum Mechanics Historical Analysis

and Open Questions. 1995 ISBN 0-7923-3480-972. A. Peres: Quantum Theory: Concepts and Methods. 1995 (see for hardback edition, Vol. 57)

ISBN Pb 0-7923-3632-173. M. Ferrero and A. van der Merwe (eds.): Fundamental Problems in Quantum Physics. 1995

ISBN 0-7923-3670-474. F.E. Schroeck, Jr.: Quantum Mechanics on Phase Space. 1996 ISBN 0-7923-3794-875. L. de la Pena and A.M. Cetto: The Quantum Dice. An Introduction to Stochastic Electrody-

namics. 1996 ISBN 0-7923-3818-976. P.L. Antonelli and R. Miron (eds.): Lagrange and Finsler Geometry. Applications to Physics

and Biology. 1996 ISBN 0-7923-3873-177. M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers: The Enigmatic Photon. Volume 3: Theory and

Practice of the B(3) Field. 1996 ISBN 0-7923-4044-278. W.G.V. Rosser: Interpretation of Classical Electromagnetism. 1996 ISBN 0-7923-4187-279. K.M. Hanson and R.N. Silver (eds.): Maximum Entropy and Bayesian Methods. 1996

ISBN 0-7923-4311-580. S. Jeffers, S. Roy, J.-P. Vigier and G. Hunter (eds.): The Present Status of the Quantum Theory

of Light. Proceedings of a Symposium in Honour of Jean-Pierre Vigier. 1997ISBN 0-7923-4337-9

81. M. Ferrero and A. van der Merwe (eds.): New Developments on Fundamental Problems inQuantum Physics. 1997 ISBN 0-7923-4374-3

82. R. Miron: The Geometry of Higher-Order Lagrange Spaces. Applications to Mechanics andPhysics. 1997 ISBN 0-7923-4393-X

83. T. Hakioglu and A.S. Shumovsky (eds.): Quantum Optics and the Spectroscopy of Solids.Concepts and Advances. 1997 ISBN 0-7923-4414-6

84. A. Sitenko and V. Tartakovskii: Theory of Nucleus. Nuclear Structure and Nuclear Interaction.1997 ISBN 0-7923-4423-5

85. G. Esposito, A.Yu. Kamenshchik and G. Pollifrone: Euclidean Quantum Gravity on Manifoldswith Boundary. 1997 ISBN 0-7923-4472-3

86. R.S. Ingarden, A. Kossakowski and M. Ohya: Information Dynamics and Open Systems.Classical and Quantum Approach. 1997 ISBN 0-7923-4473-1

87. K. Nakamura: Quantum versus Chaos. Questions Emerging from Mesoscopic Cosmos. 1997ISBN 0-7923-4557-6

88. B.R. Iyer and C.V. Vishveshwara (eds.): Geometry, Fields and Cosmology. Techniques andApplications. 1997 ISBN 0-7923-4725-0

89. G.A. Martynov: Classical Statistical Mechanics. 1997 ISBN 0-7923-4774-990. M.W. Evans, J.-P. Vigier, S. Roy and G. Hunter (eds.): The Enigmatic Photon. Volume 4: New

Directions. 1998 ISBN 0-7923-4826-591. M. Redei: Quantum Logic in Algebraic Approach. 1998 ISBN 0-7923-4903-292. S. Roy: Statistical Geometry and Applications to Microphysics and Cosmology. 1998

ISBN 0-7923-4907-593. B.C. Eu: Nonequilibrium Statistical Mechanics. Ensembled Method. 1998

ISBN 0-7923-4980-6


94. V. Dietrich, K. Habetha and G. Jank (eds.): Clifford Algebras and Their Application in Math-ematical Physics. Aachen 1996. 1998 ISBN 0-7923-5037-5

95. J.P. Blaizot, X. Campi and M. Ploszajczak (eds.): Nuclear Matter in Different Phases andTransitions. 1999 ISBN 0-7923-5660-8

96. V.P. Frolov and I.D. Novikov: Black Hole Physics. Basic Concepts and New Developments.1998 ISBN 0-7923-5145-2; Pb 0-7923-5146

97. G. Hunter, S. Jeffers and J-P. Vigier (eds.): Causality and Locality in Modern Physics. 1998ISBN 0-7923-5227-0

98. G.J. Erickson, J.T. Rychert and C.R. Smith (eds.): Maximum Entropy and Bayesian Methods.1998 ISBN 0-7923-5047-2

99. D. Hestenes: New Foundations for Classical Mechanics (Second Edition). 1999ISBN 0-7923-5302-1; Pb ISBN 0-7923-5514-8

100. B.R. Iyer and B. Bhawal (eds.): Black Holes, Gravitational Radiation and the Universe. Essaysin Honor of C. V. Vishveshwara. 1999 ISBN 0-7923-5308-0

101. P.L. Antonelli and T.J. Zastawniak: Fundamentals of Finslerian Diffusion with Applications.1998 ISBN 0-7923-5511-3

102. H. Atmanspacher, A. Amann and U. Muller-Herold: On Quanta, Mind and Matter Hans Primasin Context. 1999 ISBN 0-7923-5696-9

103. M.A. Trump and W.C. Schieve: Classical Relativistic Many-Body Dynamics. 1999ISBN 0-7923-5737-X

104. A.I. Maimistov and A.M. Basharov: Nonlinear Optical Waves. 1999ISBN 0-7923-5752-3

105. W. von der Linden, V. Dose, R. Fischer and R. Preuss (eds.): Maximum Entropy and BayesianMethods Garching, Germany 1998. 1999 ISBN 0-7923-5766-3

106. M.W. Evans: The Enigmatic Photon Volume 5: O(3) Electrodynamics. 1999ISBN 0-7923-5792-2

107. G.N. Afanasiev: Topological Effects in Quantum Mecvhanics. 1999 ISBN 0-7923-5800-7108. V. Devanathan: Angular Momentum Techniques in Quantum Mechanics. 1999

ISBN 0-7923-5866-X109. P.L. Antonelli (ed.): Finslerian Geometries A Meeting of Minds. 1999 ISBN 0-7923-6115-6110. M.B. Mensky: Quantum Measurements and Decoherence Models and Phenomenology. 2000

ISBN 0-7923-6227-6111. B. Coecke, D. Moore and A. Wilce (eds.): Current Research in Operation Quantum Logic.

Algebras, Categories, Languages. 2000 ISBN 0-7923-6258-6112. G. Jumarie: Maximum Entropy, Information Without Probability and Complex Fractals. Clas-

sical and Quantum Approach. 2000 ISBN 0-7923-6330-2113. B. Fain: Irreversibilities in Quantum Mechanics. 2000 ISBN 0-7923-6581-X114. T. Borne, G. Lochak and H. Stumpf: Nonperturbative Quantum Field Theory and the Structure

of Matter. 2001 ISBN 0-7923-6803-7115. J. Keller: Theory of the Electron. A Theory of Matter from START. 2001

ISBN 0-7923-6819-3116. M. Rivas: Kinematical Theory of Spinning Particles. Classical and Quantum Mechanical

Formalism of Elementary Particles. 2001 ISBN 0-7923-6824-X117. A.A. Ungar: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession. The

Theory of Gyrogroups and Gyrovector Spaces. 2001 ISBN 0-7923-6909-2118. R. Miron, D. Hrimiuc, H. Shimada and S.V. Sabau: The Geometry of Hamilton and Lagrange

Spaces. 2001 ISBN 0-7923-6926-2


119. M. Pavsic: The Landscape of Theoretical Physics: A Global View. From Point Particles to theBrane World and Beyond in Search of a Unifying Principle. 2001 ISBN 0-7923-7006-6

120. R.M. Santilli: Foundations of Hadronic Chemistry. With Applications to New Clean Energiesand Fuels. 2001 ISBN 1-4020-0087-1

121. S. Fujita and S. Godoy: Theory of High Temperature Superconductivity. 2001ISBN 1-4020-0149-5

122. R. Luzzi, A.R. Vasconcellos and J. Galvao Ramos: Predictive Statitical Mechanics. A Nonequi-librium Ensemble Formalism. 2002 ISBN 1-4020-0482-6

123. V.V. Kulish: Hierarchical Methods. Hierarchy and Hierarchical Asymptotic Methods in Elec-trodynamics, Volume 1. 2002 ISBN 1-4020-0757-4; Set: 1-4020-0758-2

124. B.C. Eu: Generalized Thermodynamics. Thermodynamics of Irreversible Processesand Generalized Hydrodynamics. 2002 ISBN 1-4020-0788-4

125. A. Mourachkine: High-Temperature Superconductivity in Cuprates. The Nonlinear Mechanismand Tunneling Measurements. 2002 ISBN 1-4020-0810-4

126. R.L. Amoroso, G. Hunter, M. Kafatos and J.-P. Vigier (eds.): Gravitation and Cosmology:From the Hubble Radius to the Planck Scale. Proceedings of a Symposium in Honour of the80th Birthday of Jean-Pierre Vigier. 2002 ISBN 1-4020-0885-6

127. W.M. de Muynck: Foundations of Quantum Mechanics, an Empiricist Approach. 2002ISBN 1-4020-0932-1

128. V.V. Kulish: Hierarchical Methods. Undulative Electrodynamical Systems, Volume 2. 2002ISBN 1-4020-0968-2; Set: 1-4020-0758-2

129. M. Mugur-Schachter and A. van der Merwe (eds.): Quantum Mechanics, Mathematics, Cog-nition and Action. Proposals for a Formalized Epistemology. 2002 ISBN 1-4020-1120-2

130. P. Bandyopadhyay: Geometry, Topology and Quantum Field Theory. 2003ISBN 1-4020-1414-7

131. V. Garzo and A. Santos: Kinetic Theory of Gases in Shear Flows. Nonlinear Transport. 2003ISBN 1-4020-1436-8

132. R. Miron: The Geometry of Higher-Order Hamilton Spaces. Applications to HamiltonianMechanics. 2003 ISBN 1-4020-1574-7

133. S. Esposito, E. Majorana Jr., A. van der Merwe and E. Recami (eds.): Ettore Majorana: Noteson Theoretical Physics. 2003 ISBN 1-4020-1649-2

134. J. Hamhalter. Quantum Measure Theory. 2003 ISBN 1-4020-1714-6135. G. Rizzi and M.L. Ruggiero: Relativity in Rotating Frames. Relativistic Physics in Rotating

Reference Frames. 2004 ISBN 1-4020-1805-3136. L. Kantorovich: Quantum Theory of the Solid State: an Introduction. 2004

ISBN 1-4020-1821-5137. A. Ghatak and S. Lokanathan: Quantum Mechanics: Theory and Applications. 2004

ISBN 1-4020-1850-9138. A. Khrennikov: Information Dynamics in Cognitive, Psychological, Social, and Anomalous

Phenomena. 2004 ISBN 1-4020-1868-1139. V. Faraoni: Cosmology in Scalar-Tensor Gravity. 2004 ISBN 1-4020-1988-2140. P.P. Teodorescu and N.-A. P. Nicorovici: Applications of the Theory of Groups in Mechanics

and Physics. 2004 ISBN 1-4020-2046-5141. G. Munteanu: Complex Spaces in Finsler, Lagrange and Hamilton Geometries. 2004

ISBN 1-4020-2205-0


142. G.N. Afanasiev: Vavilov-Cherenkov and Synchrotron Radiation. Foundations and Applications.2004 ISBN 1-4020-2410-X

143. L. Munteanu and S. Donescu: Introduction to Soliton Theory: Applications to Mechanics. 2004ISBN 1-4020-2576-9

144. M.Yu. Khlopov and S.G. Rubin: Cosmological Pattern of Microphysics in the InflationaryUniverse. 2004 ISBN 1-4020-2649-8

145. J. Vanderlinde: Classical Electromagnetic Theory. 2004 ISBN 1-4020-2699-4146. V. Capek and D.P. Sheehan: Challenges to the Second Law of Thermodynamics. Theory and

Experiment. 2005 ISBN 1-4020-3015-0147. B.G. Sidharth: The Universe of Fluctuations. The Architecture of Spacetime and the Universe.

2005 ISBN 1-4020-3785-6148. R.W. Carroll: Fluctuations, Information, Gravity and the Quantum Potential. 2005

ISBN 1-4020-4003-2

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